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JOURNAL OF RESEARCH of the National Bureau ot Standards— D. Radio Propagation 
Vol. 64D, No. 2, March-April 1960 

Terrestrial Propagation of Very-Low-Frequency 

Radio Waves 

A Theoretical Investigation 
James R. Wait* 

(September 11, 1959) 

A self-contained treatment of the waveguide mode theory of the propagation of very- 
low-frequency radio waves is presented. The model of a flat earth with a sharply bounded 
and homogeneous ionosphere is treated for both vertical and horizontal dipole excitation. 
The properties of the modes are discussed in considerable detail. 

The influence of earth curvature is also considered by reformulating the problem using 
spherical wave functions of complex order. The modes in such a curved guide are investi- 
gated and despite the initial complexity of the general solution, many interesting and 
limiting cases may be treated in simple fashion to yield useful and convenient formulas for 
calculation. 

Other factors considered are the influence of the earth's magnetic field, antipodal effects, 
resonator type oscillations, and the influence of stratification at the lower edge of the 
ionosphere. 

1. Introduction 

The concept that radio waves are channeled between the earth and the ionosphere as in 
a waveguide has proved to be very useful at very low frequencies «30 kc). in 1919, G. N. 
Watson [1] * employed this approach when he considered, at least in a formal way, the propaga- 
tion of electromagnetic waves between an idealized homogeneous spherical earth and a concentric 
reflecting layer. Because of the extremely poor convergence of the exact series solutions, 
Watson devised a technique to convert this to a more rapidly converging series using function- 
theoretic means. The Dew representation corresponds to the sum of residues at poles in the 
complex plane and hence 4 the name "residue series." The waves associated with these poles 
are the waveguide modes. Watson studied the numerical properties of these modes for the 
case of long waves or low frequencies and on assumption of a very highly conducting shell. 
This particular aspect of his investigation was prompted by the recent discoveries of Marconi 
that radio waves decay much more slowly with distance than predicted on the basis of classical 
diffraction theory in the absence of a reflecting shell. Watson found that the modes of low 
attenuation behaved like 

g—a^f/ad 



(sin d/a) * 

where d is the great circle distance, /is the frequency, a is the conductivity of the reflecting 
ionosphere, a is the radius of the earth, and a is a constant. For frequencies in the range from 
about 20 to 40 kc, observed field strengths behaved more or less in this fashion if the effective 
ionospheric conductivity was taken to be about 10~ 4 mhos/m or a conductivity of the same order 
as "tap water". Actually for frequencies in this range some 10 to 30 modes would be excited 
and if the complete mode sum were considered, the calculated field strength versus distance 
curve using such a model would show many rapid and violent undulations. Such a behavior 
is not observed under normal conditions and this fact alone is sufficient cause to reject this 
model even from a phenomenological viewpoint. The same model with certain refinements 
has been discussed more recently by Rydbeck [2] in a monograph, Bremmer [3] in his book, 
Schumann [4 to 6 J in a series of papers, and most recently by Kaden [7], From the frequency 



* Central Radio Propagation Laboratory, National Bureau of Standards, Boulder, Colo. 
1 Figures in brackets indicate the literature references at the end of this paper. 

153 



analysis of atmospheric wave forms [8, 9], it is known that the attenuation rate does not vary 
like -yff except possibly at frequencies near 1 kc. Actually, the attenuation rate decreases 
with increasing frequency in the range from about 2 to 18 kc and thereafter increases. A 
behavior of this kind is highly suggestive of a Brewster angle effect. Such a proposal was 
first made by Namba [10] as far as this writer can ascertain. It thus appears than the iono- 
sphere at vlf is behaving more like a magnetic wall (tangential H near zero) rather than an 
electric wall (tangential .Enear zero) as postulated by Watson, Bremmer, and Schumann. 

Contributions to the waveguide mode concept have also been made by Budden [11 to 13 
inch] who unlike the early workers did not assume a highly conducting reflective layer at the 
outset. His model was a vertical electric dipole source in the space between the surface of a 
flat perfectly conducting ground and a sharply bounded homogeneous ionosphere. Various 
extensions and generalizations have been made by AFpert [14, 15], Lieberman [16], Wait 
[17 to 22 inch], Howe [23], and Friedman [24]. Their work is referred to from time to time in 
the text. It is the purpose of the present paper to present a unified treatment of the mode 
theory of vlf propagation. The results include much of the above work as special cases. In 
fact, in some instances, the analysis follows the work of Budden, Bremmer, and Rydbeck rather 
closely, although many new results are derived. While no attempt is made to present numerical 
results, limiting cases and simplified forms of the general solutions are discussed in some detail. 
An extensive bibliography is included in this paper for the convenience of those who are more 
interested in related numerical results and experimental data. It is intended that this paper 
will serve as a theoretical basis for subsequent papers by A. G. Jean, W. L. Taylor, A. D. Watt, 
and the author. 

2. Basic Concepts 

To introduce the subject a very simple model is chosen. The earth and the ionosphere 
are represented by perfectly conducting planes. In terms of a cylindrical coordinate system 
(p, <£, z) the ground surface is the plane 2—0 and the lower boundary of the ionosphere is the 
plane z=h. The source is now considered to be a vertical electric dipole located on the ground. 
The electric field observed at some other point on the ground plane has only a vertical compo- 
nent and can be deduced by considering the images of the source dipole. These images are 
located at z=±2h, ±4A, ±6/i, etc., and all have equal sign and magnitude, because of the 
assumed perfect conductivity of the bounding walls. These images will always direct a wave 
broadside since the radiation from each image is in phase. At a distance which is large com- 
pared to A, this field can be calculated by replacing the line of dipole images by a continuous 
line source carrying an equivalent uniform current I a . This current is the average current 
along the z axis and is given by 

j ds 

Ia h l 

in terms of the height ds of the dipole and its current /. Now the field of a line source of cur- 
rent is well known and thus 

E z J-f^ Hp {kp) =^p H^ (kp) (2.1) 

where H^ 2) (kp) is the Hankel function of the second kind of argument kp, p. is the permeability 
of the space, oo is the angular frequency, and k = 2-ir/\. When p^> X, the Hankel function can 
be replaced by the first term of its asymptotic expansion and this leads readily to 

E *~%T^ eiT/ * e ~ ikp < 2 ' 2 > 

1 h(Xp) 2 

where ry=( i u/e)i^l207r. As mentioned above, this field corresponds to the radiation directed 
broadside so the rays are parallel to the bounding walls. However, there will be other angles 
where the rays emanating from each of the dipoles in the line of images are also in phase. 

154 



Such a resonance condition exists when 

2hC=n\ (2.3) 

where C is the cosine of the angle subtended by the rays and the 2 axis and n is an integer 
(see fig. 1). It is seen that for each value of n there are two families of rays which have the 
same radial phase velocity (i.e., = c/S) but with opposing vertical phase velocities (i.e.,= 
±c/C). Again the radiation of these sets of waves (i.e., modes) can be imagined to originate 
from an equivalent line source. The strength of this line source is IS where 8 is the sine of the 
angle subtended by the rays and the vertical direction. To obtain the resultant vertical field, 
this must be again multiplied by S. Consequently the resultant field of all the families of 
rays or modes is obtained by summing over integral values of n from to 00 to give 

E _mlte £ ^ slR{f {kSnp) (2 4) 

where € =1, e n =2(n=l, 2, 3 . . . ) and S n =(l — CI)* and C n =n\/2h. The term n=0, corres- 
ponding to mode zero discussed above, is only included once in the summation, whereas the 
higher modes are included twice. In the far field, this expression for the field reads 

E *~Jv^ eiv/ * S enSVe-^np. (2.5) 

2h(\p)* «=o,i,2, ... 




=h 



z = o 



arc cos C 



(image) 



Figure 1. Depicting ray-geometry corresponding to the first 
mode between parallel plates; for resonance, X = 2/iCi. 

Up to this point the bounding walls have been assumed to be of perfect conductivity: 
The reflection coefficients for the rays are always +1. Another simple case is when the upper 
boundary has a reflection coefficient of — 1 corresponding to a perfect magnetic conductor, and 
the lower boundary still has a reflection coefficient of + 1 corresponding to the more common 
perfect electrical conductor. For this situation the images are now located at z= ± h, ±2h, 
. . . but now they alternate in sign. It may be observed that there is no coherent family of 
rays directed broadside. This would have been the zero-order mode. The resonance condition 
for the modes is now 

2hC=(n-%)\ (2.6) 

where n=l, 2, 3, ... . The corresponding expression for the vertical electric field is thus 
given by 

Ez=z ^Ids £ S 2 H w {kSnp) (2.7) 

«l ?!= 1,2,3, . . 

155 



where the summation starts at n=l and includes all positive integers. In the far field 

E *~j!7^ eiT/i ^ Sl^e-^no. (2.8) 

tl(\p) /2 72=1,2,3, . . . 

In the foregoing discussion the observer and the source, which is a vertical electric dipole, 
are located on the ground plane. The above results are easily generalized to a finite source 
height, z j and a finite observer height, z, by inserting the factor cos (kz C n ) cos (kzC n ) inside 
the summations of eqs (4), (5), (7), and (8). This can be verified by returning to the image 
picture and noting now that they are located at z=—h, ±(2h J r z ), ±(2h—z ), ±(ih J rz ) f 
±(4h—z ), .... It may also be observed that the cos (kzC n ) when replaced by [exp(ikzC n ) 
+ exp (—ikzC n )]/2 can be identified as a family of upgoing and downcoming rays within 
the guide. 

The important modifications of the preceding formulas as a result of imperfect reflection 
can be obtained by a rather simple physical argument. The complete treatment requires a 
more mathematical approach which is to be described later on. 

The reflection coefficient for a ray incident on the ground plane at an angle (whose cosine 
is C) is denoted R g (C). The corresponding reflection coefficient for the upper boundary which 
is the lower edge of ionosphere is denoted Ri(C). The resonance condition now has the form 

R g (C)Ri(0)e~ i2khc =e- i27rn (2.9) 

which reduces to eq (3) if the reflection coefficients are both +1 and reduces to eq (6) if one 
reflection coefficient is +1 and the other — 1. Physically, the above more general form can 
be the condition for a ray to traverse the guide twice, be reflected at each boundary, and yet 
still suffer a net phase shift of 2 w n radians where n is an integer. Since R g (C) and Ri{C) may 
be complex and less than unity the value of C (i.e., C n ) which satisfies the resonance equation 
may also be complex. The angle of incidence of these rays in the guide are thus also complex. 
The corresponding value of S n is also complex and this results in attenuation of the wave in 
the radial direction. In fact, the attenuation constant is minus the imaginary part of kS n in 
nepers per unit distance. 

When the angle or its cosine C must be complex in order to satisfy a resonance equation, 
the resulting waves are damped. The numerical solution of such a complex resonance 
equation is quite difficult, in general, since it is not usually possible to obtain an explicit expres- 
sion for C in terms of known parameters. This aspect of the problem is discussed in a later 
section. 

3. Formulation for Flat Earth Case 

3.1. Vertical Dipole Excitation 

It is now desirable to formulate the problem in a more definite fashion. A vertical electric 
dipole of moment / ds is placed in a homogeneous plane layer bounded by two plane interfaces 
(see fig. 2). The lower interface is at z=0 corresponding to the surface of a homogeneous 
ground of conductivity a g and dielectric constant e g . The upper interface at z=h is the lower 
edge of a homogeneous ionosphere which for the moment is assumed to be isotropic and has 
effective electrical constants <r t and e t . The fields in these regions can be derived from a Hertz 
vector which has only a z component, II z . Thus, for h^z^O 



(3.1) 



z^dpte 11 * 




H,=0 


E*=0 




rr ■ ™> 

HA,=—iea)^— 
dp 


E,=(k*+^ 


u 


H z =0 



where &=(e J u)2a>=27r/\. 



156 



z =h / / / / / / 



( ionosphere) 
0-\ €j Ho 



z = o -^ 



=///-/// r//. r /.', 



(air) 
*o ho 



(ground) 



^g ^g Mo 

Figure 2. Cylindrical coordinate system for the vertical dipole 
between the two plane interfaces. 



Similar expressions are applicable for the regions 2>A and 2<0 where the z directed Hertz 
vectors are denoted n^ and U { z g) , respectively, and the corresponding wave numbers are k t and 
kg, respectively. The formal solution of this problem is obtained in a straightforward fashion 
by requiring that the tangential field components E p and H+ are continuous across the two plane 
interfaces. An equivalent statement of these matching conditions is as follows: 



k 2 U 2 =k 2 g U z (gy 

ML m z (g) 



dz 



dz 



k 2 Tl r - 


=*?ni <)- 


dz = 


" 52 _ 



for 3=0, 



for z = h. 



(3.2) 



(3.3) 



To facilitate the solution, the primary excitation resulting from the source dipole is repre- 
sented as a spectrum of plane waves. This well-known representation, for the primary Hertz 
function, is given as follows: [25] 



II 



(p). 



M exp {-ik[p 2 +(z~z ) 2 ]i} = Mik 
[p 2 +(s- S „) 2 ]* : 2 



j> 



(kS P ) exp [-ikC\z- 



\dC 



(3.4) 



where M=I ds/(4wio)e) and S=(l — C 2 p. The integration variable can be regarded as the 
cosine of the angle of incidence of the plane waves in the spectrum, r is the contour of integra- 
tion and it extends from — oo along the negative real axis to the origin, then out along the real 
axis to +00. It should be noted since C can be greater than unity complex angles in the 
spectrum occur. The above form for the primary excitation then suggests that the resultant 
Hertz function for the three regions can be written in the respective forms 



(I) n z =n<* ) + f [A (C)e~ ikCs +B (C) e +ikcz ] H^ kSp) dC 

for Q^z^h; 



(II) 

for 2^0; and 

an) 



ni*>= { G(C)e +lk f'*H!?>{kSp)dC 
If <« = [ I(C)e~ <W#<» (kS P )dC 



(3.5) 



(3.6) 



(3.7) 



for zy>h. In the above, it can be verified that these Hertz functions satisfy the appropriate 
wave equation subject to the conditions that 



157 



(3.8) 



where 



N z^+i^y and N (°i+iwy . 

g \ leu / V lew J 



Terms containing exp (— ik g C g z) and exp (ik l C i z) are not permitted since they would violate 
the Sommerfeld radiation condition at |2|->». 

The form of the unknown function A(C), B(C), 0(C), and 1(C) can be obtained explicitly 
by using the four equations of continuity. This purely algebraic process is easily carried out 
and further details are omitted. The resultant Hertz function for the air region is explicitly 
given by 

TI t =~j r F(C)H^ (kS P )dC (3.9) 

where 

(e iWz + R g e- ikCz ) (e ikc(h - g o> + R^-^- z ^) , 

(C) " e iWh (l-R g R i e~ 2ikW ) [ } 

and 

AT P—P 
Br=B t (C)= ^ + ^ (3.11) 

AT P—P 

It can be immediately noted that the integrand has poles where 

This is the (complex) resonance equation obtained in the previous section by intuitive reasoning. 
The integral may be evaluated by using function- theoretic means. The contour is trans- 
formed to the S plane. Thus eq (9) becomes 

u _ikM C F{C)H i 2) {kSp) §d£ (3 13) 

where the contour r may now be taken as the real axis from — <» to + °° in the S plane. 

The contour is now closed by semicircles in the lower half -plane as indicated in figure 3. 
Because of the branch point at S= + 1 and its associated branch line drawn vertically downward, 
the closing contour runs from one Riemann sheet to the other in the manner indicated. After 
making two circuits the contour closes on itself. The contours are indented at other branch 
points in the manner shown for B on the figure. These branch points are located well below 
the real axis (i.e., imaginary part of £>>1) and the corresponding branch cut integrations lead 
to waves which are heavily damped provided 

and£ P |Al-l]»l. (3.14) 

Now the line integral around the complete circuit in the two sheeted Riemann surface is equal 
to — 47ri times the sum of the residues of the poles of the integrand. The poles which occur in 
pairs are located on both Riemann sheets. For highly conducting walls, a number (at least 
one) is located just below the real axis between the origin and the branch point at -f 1. The 
remainder are located along or near the negative imaginary axis. The contribution from these 
latter poles is very small and they correspond to the waveguide modes beyond "cut-off." 

The contribution along the semicircles is seen to vanish if the radius R approaches infinity. 
This is assured by the presence of the Hankel function Ho 2) (kSp) which is exponentially de- 
creasing in the lower half-plane of S. Consequently, each of the two integrations along the 

158 




Figure 3. S 'plane. 
Branch point; x, poles. 



real axis are approximately equal to — 2t% times the sum of the residues at S=S n . When the 
integral is expressed in the original C plane the residue series may be regarded as the contri- 
butions from the poles at C=C n which for n=Q, 1, 2, . . . are in the first quadrant and for 
n= — l, — 2, — 3, . . . are in the third quadrant. This leads to 



U z = wkM^2 



1 






H?> (kS„ P ) 



(3.15) 



where the square bracket term is the residue of the function /''('') at the pole C=C n . Carrying 
out the differentiation and making use of the resonance condition 



leads readily to 



where 



R t (C n )B t {C n )e- 



?»=1 



n ^ ^g H n ){kSnp) y n(go) fn(z) 5n{n 

lb n = — co 



r wt t (C)ii g (C)]/i>c -\- 1 

" ( ' L 2khR i (C)R K (C) 

o=c n 



where 



/.(«o 



e ttc n l +R g (C n )e- ,kC «' 



(3.16) 
(3.17) 

(3.18) 
(3.19) 



2[R g {C n )V 

and/„(2 ) has exactly the same form. 

When the walls are perfectly conducting J B i ((7)=i?y(C0 = l,the factor 8 n (C) becomes unity if 
. and becomes % if n = 0, and/ n (s)=cos kC n z. The above expression can then 



w=l, 2, 3 



y *"y ^j 



be written 



iirlVf °° 

n z = ~ S tnHr (kS nP ) cos (A:C„2o) cos (kC n z) 

fl n=0,l,2, . . . 



(3.20) 



where e =l, e n =2(n? £ 0). The corresponding value of the electric field component E z can be 
expressed 

,=(f+|J)iI,=-^^ f) e »SSff«'(fcS,p)co 8 (kC n z ) cos (kC n z) (3.21) 



#,= 



4A 



w=0,l,2, 



which is in agreement with eq (2.4) obtained from physical intuitive reasoning. 

The extension to the case when the source is a vertical magnetic dipole is simple. Formally 
the above results are still valid if I is replaced by the magnetic current K. E z then becomes H z 
and the field is essentially horizontally polarized. The reflection coefficients R g (C) and Rt(C) 
are to be replaced by their counterparts for horizontal polarization. 



159 



Explicit^ these are given by 

C-N t C t 

''C+N.C, 
and 



Ri(0 = n , Zn (3-22) 



E w=wB; . (3 - 23) 

It may be noted, since R( — C) R(C) = 1 where R is an}^ of the reflection coefficients R iy 
R g , Rt, or R h g , that for negative values of n 

^-(n4-l)* == ^ n- 

-(-00 c» 

It then follows that the summations S • • • may be replaced by 2 Xj • • • everywhere 

— oo " 

since 8 n (C) = 8 n (—C). 

For convenience in numerical computation, it is convenient to express the field components 
as a ratio to the quantity 

E =i(r}l*)Ids(e- ik »)/p (^120tt). (3.24) 

E is the field of the source at a distance p on a perfectly conducting ground. Thus for both the 
source and the observer near the ground it is not difficult to show by means of eqs (1), (17), 
and (19) that 



where 



where 

and 
where 



E z =WE 



W ^-iir £ e ik > S hSim» (kS nP ) (3.25) 

Ep =SEq 



S et£i« a 'j:dJ3JBp(kS 1 u>) (3.26) 



H^TE /r, 



T^-tt £ e**> S 6.S n Hl*> (kS nP ). (3.27) 



In the above it has been assumed that |A^| 2 >>1. 

When lcp^>^>lj corresponding to the "far-zone," the above expressions may be simplified 
since the Hankel functions may be replaced by the first term of their asymptotic expansion. 2 



2 The relevant expansions are 



wK£)*~S[i-£+5fe---] 

„ ? , (T)s (|)» e -^[ 1+i |__H_...]_ 

160 



This leads to the compact result 
'W 



s 

T 



(pA)!/[x-t] 

(A/X) 



2«, 

n = 



-Sl/N g 

si 



g-i'2TrS n p/\ 



(3.28) 



which is valid for p>>\. As expected, the ratio of W to T for a given mode is S n which fol- 
low order or grazing modes is of the order of unity. The ratio of S to T for a given mode is 
~l/Ng which is ver}^ small compared to unity; in fact, it vanishes for a perfectly conducting 
ground as it must. 

3.2. Horizontal Dipole Excitation 

The previous section contains the formulation for a vertical dipole source. The corres- 
ponding treatment for a horizontal dipole is also quite straightforward although the lack of 
symmetry increases the complexity. Often in the 4 radio propagation literature the statement 
is made that the fields of a horizontal electric dipole are the same as, or proportional to, the fields 
of a vertical magnetic dipole at the same location. This is only true broadside to the horizontal 
dipole where the field is purely TE (transverse electric) or horizontally polarized. For other 
directions, the field has a TM (transverse magnetic) component corresponding to vertical 
polarization. The modes corresponding to the TM waves may have much smaller attenuation 
than the modes of the TE type and thus it is desirable to formulate the problem directly with 
a horizontal dipole source. 

As in the previous section the earth and the ionosphere are assumed to be bounded by 
parallel planes separated by a distance h. Choosing a rectangular coordinate system {x } y, z), 
the dipole is located at z—z Q and is parallel to the x axis (see fig. 4). 

The solution for a horizontal dipole over a homogeneous flat earth with no ionosphere 
(i.e., h-^co) was obtained by Sommerfeld many 3^ears ago. The generalization for the two 
interfaces is quite straightforward. A Hertz vector is introduced which has both an x component 
U x and a z component IT 2 . The fields in terms of these are 



_d ran, , dn/1 

"~ by Idx^dz J 



H x =leo} -rr — > 
by 






H z 



■ I eoo 



dy 



(3.29) 



As before a subscript g or i is added to these quantities when specific reference is made to the 
ground or the ionosphere, respectively. 

The boundary conditions at the interfaces z=Q and z = h are that tangential components 
of the fields are continuous. This, in turn, requires that k 2 U x , dU x /dx J r()II z /dz y ieull 2 and 
iewdn x /()z are each continuous at these interfaces. Integral representations of IT X and n 2 which 

161 





z 










a , € , Mo 






z~h - 


/////// 


/ / / / / / / / / / / / ,- / 










__ € o Mo 






2=0 7 














^g € q H-o 







Figure 4. Rectangular coordinate system for the horizontal 
dipole between two plane interfaces. 



are suitable for matching are 

n z =M^-+ [U(C)e-"< c *+V(Oe+ ikCz ]H a ™(kSp)dC (3.30) 

n 2 =^ (* {X(C)e- ikCz +Y(C)e+ ikC W 2) (kSp)dC (3.31) 

for O'Cs'CA.. Similar expressions are used for the x and z components of the Hertz vector in 
the spaces s<C0 and z^>h. Applying the boundary conditions involving II ^ only, leads directly 
to the following solutions for the unknown coefficients in n^: 

r R\+R\m exp [-2ikC(h-z )} l , 

U(C)-^ i-E^RUxp (~2ikCh) J eXp( lkt3o) {6 - 32) 

^ g )-[ ?-|^?(-2^) 0] ] -P [" ^-.)] (3.33) 

where J?J and 7?? are the complex reflection coefficients given by eqs (22) and (23) and they 
are also functions of C. 

The remaining two boundary conditions, namely, the continuity of ieo)H z and dU z /dz 

+ bU x /dx enable the coefficients X(C) and T(C) to be found in terms of U(C) and V(C). The 
connecting relations are 

P-QB g exp (-ikCh) . 

^ L) -l-R t R g exp (-2ikCh) (6 } 
and 

v ^ rn PRi exp (—2ikCh) — Q exp (—ikCh) , . 

r(C) -~ l-ff,fl f exp (-i2&Cft) ( ^ 5j 
where 

P=[*-«*o+tf+F] (^f) (3.36) 

It is understood that i?*, i? g are functions of C and are defined by eqs (11) and (12). 

The integral for U x can be observed to have precisely the same form as the z component of 
a (magnetic) Hertz vector for a vertical magnetic dipole. This in turn has the same general 
form as the z component of the (electric) Hertz vector for a vertical electric dipole. The 
residue series representation for Jl x is given by 

n«s^S #o 2) {kS m p)fi(zo)Ji{z)^{C) (3.38) 



and 



where 

tt(O-[i + i '«' c W W c ] 



162 



where the summation is over the poles of the integrand at C=C m of the integral in eq (30). 
These are solutions of 

R h i (C)R h g (C)e- 2ikhc =e- i2 ™ (3.39) 

for integral values of m. 

The height functions have the form 

2/,;K-)-[^(^ r )J~^ + ^+[/^(0]^-^ (3.40) 

Of particular interest is the vertical magnetic field component. It is given by 

TT . dU x . dU x . 

H 2 =—lea3-^ — = — leu — — sin 
dy dp 

Tfhk 
=sin <A if S S m H? (kS m p)f»(2om(z)dUC). (3.41) 

WhenlfcS^pj^^ 1 or when p^> ^> X, the first term of tin 4 asymptotic expansion of the Hankel 
function IIi 2) (kS„,p) may be employed. 'Phis leads to 

ff,«— ^^*^ES£/.. foO/. (z)e^-^ 5* (C) . (3.42) 

The other field components involve integrals which may be treated in the same way. 
Also of great interest is the vertical electric field. It is not difficult to show that 

E t = ~ [^ly/o^ (kS nP ) g n (z )f n (z) S n (C)] (3.4:0 

where 

2sr»(2o) = C U [R,{CJ]-* exp (*<7„2)-(7 n [#, (<?„)]* exp {-ikC n z) . (3.44) 

It may be noted that 

kg n (z,) = -i^-fn(zo)- (3.45) 

The summation is now over the roots C=C n of the equation 

R i (C)R ff (C)e- i2rkCh =e- i2irn . (3.46) 

When p^>^>\ the above expression simplifies to 

# 2 ^cos £ ^L 1__ p s!/ re (^)^(^ )^ (1 -^5 n ((7). (3.47) 

When l&C^ol^l and l&C^l^l the preceding simplifies even further to 

ftaMB^l^^SSy^HW. (3-48) 

4. Properties of the Modes for Flat Earth Case 

4.1. Vertical Polarization 

Much has been written in the literature on the numerical characteristics of the modes. 
Controversy concerning the method of numbering the modes has also arisen. It is the opinion 
that much of this discussion has been unnecessarily involved. The important thing is to sum 
over all modes which are excited by the dipole. Consequently only these modes need be 

163 



considered. Because of the form of the integration contour r the relevant solutions must 
satisfy 

R g (C)R i (C)e- 2ikhC =e- i2 ™ (4.1) 

and have their real and imaginary parts positive. That is, C n is located in the first quadrant. 
The numbering is then assigned in such a way that there is continuity in the limiting case of 
perfect conductivity (i.e., R g =Ri=l). 

irTl 
C n =TT with 7i=0,l 7 2,3, .... 

As Dr. H. H. Howe points out, this is not quite unambiguous when both the ground and 
the ionosphere are both imperfectly conducting. The more general statement of the rule 
is [23] : 

For a fixed value of kh, determine n on the assumption of perfectly conducting walls, then 
<r g and <Ji in turn are to decrease continuously to their prescribed values while C varies continu- 
ously. For walls of high but finite conductivity this means that mode has a minimum atten- 
uation and the other modes have successively higher attenuation as n increases. For poorly 
conducting walls, this is not necessarily so, and in fact, in cases of most practical interest for 
the vlf band the mode of lowest attenuation is of order one. 3 

Numerical values for C n are available and will not be quoted here. Some properties of 
the modes, however, may be simply obtained without resorting to a full numerical solution. 
For example, if the walls are highly conducting the reflection coefficients may be approximated 
as follows: 

R ^=w^ur l -wc^ exp (-m) (4 - 3) 

subject to |C| 2 >>|A^ ff | -2 and \Nf\~ 2 . Then the resonance equation is simplified to 

khC=irn J [-i -^ (4.4) 

where A = (l/N t +1/N g ). 

Regarding A/C as a small quantity, this can be solved to give 

where € =1, e n =2, (715*0). 

The magnitude of the second term must be small compared to the first term for the above 
perturbation method to be valid. This restriction and the previous one are both met if 
simultaneously 



-.— 



M|A|«land^«l-(^ 
Now for highly conducting walls a g »e g o) and o->>€jco and thus 



3 The mode numbering system described above is somewhat different from Budden. For a fixed value of <r,- he starts with a very small value 
of kh, increases it continuously and requires that C varies continuously for the same n value. [13] 

164 



Consequently 



and 



The influence of finite conductivity is thus to increase the real part of S n and consequently the 
phase velocity c/Re S n is decreased relative to the free space value c. As expected the finite 
conductivity produces damping and the resulting attenuation factor is — k Im S n in nepers 
per unit distance, to this approximation. 

The above approximate formulas for the real and imaginary parts of S n are the ones usually 
encountered. They have been quoted by Schumann [5] for example. It is not also appre- 
ciated that they arc not applicable for a mode which is near cutoff. This should be evident, 
however, from the second inequality given above. To relax this restriction, the resonance 
equation 

khC=wn+i~ (valid for \A\kh«l) (4.9) 

is solved as a quadratic in C to yield 

2 'HsH(sy +4 *B] ! «■"» 

The positive sign before the radical is chosen since it reduces to C=(wn/kh) when A approaches 
zero as it must. The corresponding form for S n is then given by 

^{'-OT'+V^w]'}' 

When n = 0, this simplifies to 

which reduces to eq (5) when n=0 and |A|<<M. Now since |A|M<<1 the radical can be 
expanded for n^>0 to yield 

The preceding discussion concerns walls which are highly conducting. The approximate 
solution obtained would indicate that the attenuation increases indefinitely as the conductivity 
of the walls decreases. Such is true as long as |A|<<1. For very poor conductivities this 
condition becomes violated. When \A\kh is of the order of unity, it is apparently necessary 
to solve the resonance equation by numerical or graphical means. This approach is described 
briefly in a later section. As it turns out, for a given value of n, the attenuation reaches a 
maximum value as |A| is continuously increased and thereafter diminishes and approaches a 
broad minimum. To illustrate this interesting phenomenon the resonance equation 

R e (C)R i (C)e- 2ikhc =e- 2 * in (4.13) 

is solved approximately under the condition that 

\N t C\«l and|iV,tf|»l. 
165 



<7 i =(l-^) i ^[iV!-l+(C B ) 2 ]4 (4.17) 



Thus 

R,(C) "-e-vfiioi and B^Oar^W (4.14) 

and therefore 

^+^+i*M7=«(n-l). (4.15) 

The zero-order approximation is obtained by replacing R t (C) by — 1 and R S {C) by +1. This 
would yield 

C=O n =(n-i)w/(kh), »=1,2,3, . . . (4.16) 

as mentioned in section 2. For the first-order perturbation 

S 2 \i 1 
o,=i i- 

and 

C=(.-0 s i 

since |A^r|>>l. With these simplifications it readily follows that 

G ~ 7r(7t ~- )+ ^k (4.18) 

kh-imim-i+(c n y]-i 

and 

S.= (1-CJ)*. 

When the upper medium is an ionized region, it is convenient to write 

It may be shown that for vlf, L is approximately real and has a magnitude of the order of 
unity. Furthermore, for a highly conducting ground 

where 

Then 

".»_i(i4)[(^.-jj* ^» 

Assuming (C n ) 2 <i<CL (which is true for low order modes), and that L is real, the real and 
imaginary parts of S n can be written 

EeS -= s - + ^ws:[ (r -'( v7; -,V) + ' 3 ] (4 - 20) 



and 

Im 



S n =— — ^ =- [~(<7J 2 fVI+4-V\ G~\ (4.21) 



^ ir(n—$)(n—l) 
kh (2A/X) 

and 

It may be observed that for a fixed value of h/\ and ^G the attenuation factor, — k Im S nj 
has a broad minimum when L=l. For L somewhat less than unity, the attenuation factor 
varies as L 2 or directly as the square root of the effective conductivity of the ionosphere. 
On the other hand, for L somewhat greater than unity the attenuation factor varies as L 2 or 
inversely as the square root of the effective conductivity. 

The excitation of the modes is proportional to the quantity 

d ' Mn) L 2khR g (C)R ( (C) J ( } 

c=c n 

When M|A|<<1, whore A=l/N,+1/N t , it follows from eqs (2) and (3) that 

^(GOfl.CGOaexp [-2A/CJ, 



and 



and thus 



9A 

[d[B f (Offi(C)l/aC]c-c«=^ exp [-2A/rV] 



*»(G->^+*4^] l - (4.23) 

Now the resonance condition states that 

ikhC n +A/C n =i2irn (4.24) 

and for n = this leads immediately to 

5 (C )=i/2 

while for ft =1,2,3, . . . 

On the other hand, if the upper medium is very poorly conducting such that 

|JV,tf„|<i 

and the lower medium is highly conducting 

iw.i»i 



it follows that 
and 

Thus 



R^CJRtiCJ ^-exp [_?^_-^-J (4.25) 

d r B.CO^COJs-^--^] R g (C n )R { (C n ). (4.26) 



w^ a C i -K(f-T^r ai (4 - 27) 



for d=1,2,3, . . . since the term in parentheses is always small compared to unity. 

167 



4.2. Horizontal Polarization 

In the case when the excitation is by a vertical magnetic dipole or horizontal electric 
dipole the modes excited may be of a transverse electric (TE) type. The appropriate modal 
equation is 

R h g (C) R\ (C) e - mhc =e- i2v ^~ 1) (4.28) 

where rn = 1,2,3, .... 

Now if \C/N g \ and |(7/JVi|<l 

R h g (C)g- exp (-2C/N g C g ) (4.29) 

and 

R\ (C) & - exp {-2CIN i C l ) ■ (4.30) 

The modal equation is thus simplified to 

c bm + 7m\ +ihkC = iirm (4 - 31) 

remembering that 

c * = V~WJ' Cs= V~Nl)"' 

and S 2 =l-0 2 . 

A first order solution is obtained by replacing S 12 in the expressions for C t and C„ by the 
zero order value, e.g., 

»-'-(S)' < 432 > 

The approximate solution of the mode equation is then given by 






where 



^[^s-'+C^T+h-'+C^T (4M) 

and S m =(l-C 2 m )$. When |JVJ| and |2V§|»l-feY 



it is seen that 
For |A^|<M, 
and 



A w ~ -— J — A . 



c 



'5('+«h) <« 5 > 

«-['-(s)T-4(s)I'-(s)T 

which is valid when the modulus of the second term is small compared to the first. 
For highly conducting walls 

A^|A| = 4(^)V(fy] (4.37) 



and therefore 



168 



and 

In summary, these are valid when 

and irrn/kh<^l. 

It is rather interesting to note that the above expressions for Re S m and Im S m are vitv 
similar to the corresponding expressions derived for Re S n and Im S n in the case of vertical 
polarization. [For example compare with eqs (7) and (8).] In the present case, of course, 
there is no zero order mode but apart from this, the perturbation term involving |A| now has 
an additional factor [(Tm/kh)] 2 which is less than unity if the mode is above "cutoff." Thus, 
everything else considered equal, the attenuation factor of the TE mode is decreased rela- 
tive to the TM mode in the earth-ionosphere waveguide with walls assumed to be of high 
conductivity. 

In the earlier notation it was convenient to represent the refractive index in the form 

Wl-l-j 

where L is a real number which may be comparable to or much less than unity. Thus 

H<SHY+w; <«°> 

The corresponding solution for the modal equation is obtained from 

When L<<1 this reduces to eq (36). 

5. Influence of Earth Curvature 

The curvature of the earth has been neglected up to this point. The problem is now for- 
mulated in terms of spherical coordinates (r, 0, <£), with the earth idealized as a homogeneous 
sphere of radius a, of conductivity a g , and dielectric constant e g . The lower edge of the as- 
sumed homogeneous ionosphere is located at r=a-\-h. The source vertical electric dipole is 
then located at r=a-\~Zo and the observer is at r=a-\-z (see fig. 5). In view of the intrinsic 
spherical symmetry the fields can be represented in terms of a single scalar function, \f/, as 
follows [26] 

(5.1) 

& < j ) =rI T =Hd= z Q 

where ri=(ji[c)i and F=co 2 (e— ia/co)^. 



532053—60- 



169 




(r,0,</>) 



Figure 5. Spherical coordinate system for vertical electric dipole 
between concentric spherical interfaces. 



As usual, the permeability is taken to be the same as free space for all the regions (/x= 
47rX10~ 7 ). A subscript g is affixed to a, e, etc., when reference is made to the ground and a 
subscript i for the ionosphere. Since ^ is a solution the wave equation appropriate for the 
regions, the solution may be represented in terms of spherical wave functions, 

a^h { V (k g r)P v (-cos d) for r<a 

[b^K'^k^+b^h^ikrW^cos B) for (a+h)>r>a 

cl»h™(k i r)P p (-co l s 0) for r>(a+h). 

In the above 



h^(kr)=(0H^f(kr) 



(5.2) 



where H$+l (kr) is the Hankel function of the first or second kind of order v+| with argu- 
ment At. P„(— cos 6) is the hypergeometric series which is a special case of the hypergeo- 
metric function F (a, j8, 7, z) namely 



r, / „\ 7-1/ 1 -, ■, 1 + COS ff\ 

P,(-cos 0)=Fl—v, r+1, 1, -^^ )• 



(5.3) 



The reason P„(— cos 6) is employed rather than P v (+cos 6) is due to the fact that \p must be 
regular on a ray 6=tt, whereas 0=0 is to contain the singularity which is the source of the 
field. Sommerfeld [26] has pointed out that 



limF 

0->O 



(—v,v 



+1,1, 



1+cos 6\ smvir 



log d 2 



(5.4) 



which illustrates the singular nature of \p along the polar axis. 

The quantity v is to be found from the boundary conditions that the fields Ee and H<f> are 
continuous at r=a and a-\-h. This, in turn, requires that (rj/r) c)(r\l/)/dr and k\j/ are contin- 
uous. Thus, four linear equations in the coefficients a^ , b^ , bf\ and c* 2) are obtained. In 
order that these yield a nontrivial solution, the four by four determinant of the coefficients 
should vanish. This requirement is explicitly given by eq (5) as follows: 

170 



nj\ pM»> (k/-) ~\ 



kjii l) (k g a) 







_/v\ prA<» (kr) l 



dr 

-lehi l) (ka) 



( V \r drh™ (kr) l 
\a+hJl dr J 

a+h 



khi° [k (a+h)] 



-M< 2 > (ka) 

(_n \V drh™ (kr) l 
\a+h/l dr J 

a+h 

*A» [k (a+h)} 



( 17, \f drh™ (k t r) l 
\a+h/i dr J 

a-\-h 



kthPVctia+h)] 



=0 



(5.5) 



Such an equation as this was obtained by G. N. Watson in 1919. To solve it for v without 
approximation does not seem to be possible, although if the general spherical Hankel func- 
tions of complex order and argument could be programed for a computer, an exact numerical 
solution might be obtained. In view of the idealizations of the model and the uncertainty 
of the effective electrical constants of the lower ionosphere, however, it does not seem war- 
ranted to expend too much effort in this direction. As is so often desirable in physical prob- 
lems, asymptotic approximations to the rigorous wave functions are introduced which greatly 
simplify the problem but at the same time lose some generality. 

The Debye-Watson representation of the Hankel functions are [26] 

when |(V+§)/&/*|<l but not near 1. Also \v-\-\\ and kr must be large compared to unity. 
The upper (and lower) signs are to be considered together. This is really a W.K.B. (Wentzel, 
Kramers, and Brillouin) approximation to the radial part of the wave equation. It is not diffi- 
cult to show that the resonance equation involving spherical Hankel functions can now be ex- 
pressed in the equivalent form 



R g R t exp S-%2 P fi- ^+^H dx\=exp (-t2m) 



(5.7) 



where n=0, 1, 2, 



R 



kg 


l~i (j,+i)£ l 

L (ka) 2 J 


*-* 


( k s af _ 


k s 


, (v+h)' 2 ' 
L (ka) 2 J 


h +k 


L (haf J 



(5.8) 



and 



kt 



(v + W 



R = L k 2 (a+h) 2 _ 



(v + h) 2 ni 



[} k 2 (a+h) 2 



" (v+i) 2 

L kVa+h) 2 J 



'+k 



(»+i) 2 



_ k\(a+h) 2 _ 



(5.9) 



The functions Ii c and R f quoted above can readily be identified as Fresnel reflection 
coefficients for complex angles of incidence cos -1 C and cos ~ l C, respectively. Furthermore 



C=[l-S 2 V where 8= 



v+h 



ka 



and 



C' = [l-(S') 2 } h where S"= 7 f^ i 



k(a+h) 



171 



The resonance equation can thus be written 

R g (C)R t (C') exp [ -i2k P \c 2 -\~ S^dz\=e-^ in (5.10) 



where 



and 



7 ^ )= Wl with ^=[i-^;J (5.1D 



^' )= w^i with c H l -^r\ ' (5 - 12) 

It can be seen that, in view of the relation 

(a+h) S'=aS, 

the resonance equation reduces to its flat earth counterpart as h/a tends to zero. In fact, it 
appears that if \C\^>y> (h/a)^ ^1/10, the effect of curvature can be disregarded. This condi- 
tion is violated for most of the numerical results given by Al'pert in the region from 15 to 30 
kc. He assumed that the modes could be calculated on the basis of a flat earth in all his 
work [14, 15]. 

The resonance equation quoted above for a curved earth is only valid if the W.K.B. or 
second-order approximations to the spherical wave functions are valid. In a later section the 
corresponding form of the mode equation based on the Airy or third-order approximation is 
developed following the work of Rydbeck. It is indicated from this more involved analysis 
that the second-order approximation is valid if 



(t)'o«i. 



As will be seen, this is met for most cases of practical interest if the frequency is less than 
about 15 kc. 

6. Mode Series for Curved Earth 

Following the suggestion of Sommerfeld, the field is written as a sum of modes. Thus 

+ (r,d)=j:D v z v (kr)P v (-cosd) (6.1) 

n 

for a<r<a+A, where 

z v {kr)=b ( v l) h i v l) (kr) + bl 2) hi 2) (kr). (6.2) 

The factor D„ is to be determined by insisting that the function \j/(r, 6) has the proper behavior 
at the source. The summation is over all integral values of n and the corresponding (complex) 
values of v are obtained from the resonance eq (5.5) as described in the previous section. 

Invoking the W.K.B. or second order approximation for the spherical wave functions, 
it follows that 

z v {kr) s const/l?,-* (C n ) expP+iife S\c\^~ SI) 1 dz~] 

+ R g i (C n ) exp [ - ikj\d+^ S2)* dz^y (6.3) 

This can be identified immediately as a combination of a downcoming and an upgoing wave. 

172 



The ratio of these two at the earth's surface (2=0) is R g (O n ). An alternate representation is 
zAkr) seonst-f #r* (<?») p|"+*fc fT r;, " + ? & *] 

+ /?} ((70 exp \-ik f*(c»+^ S») <fc~| | (6.4) 

which is a combination of an upgoing wave and a downeoming wave. The ratio of these two 
at the lower edge of the ionosphere (z=h) is li t (C n ). It should be noted that 



C 



:«(c»+f s»)* 



since (a+h) S' n =aS n and h/a « 1. The internal consistency of these two representations 
at = and h for 2„(&r) can be readily demonstrated from the relation 

\=R\{C n ) R\{C' n ) exp [-*/ o *(^+f Slf &] (6-5) 

which also indicates that the multiplicative constant is the same for the two representations. 
In what follows the constant can be absorbed into the factor I) v . 

To study the orthogonality properties of the modes, the following integral is considered 

1= Z v (p)z,(p)dp (6.6) 

J ka 

where v and \x are two sets of modes. Now quite generally the function z v {p) satisfies 

P r p (f>z,) + [p 2 -v(v+l)]z v =0 (6.7) 

and there is a similar relation for g„. These two equations are now multiplied by 2 M and z Vf 
respectively and integrated over the domain ka to kb, to obtain 



( 9 d ( 9 \ ~ d < » \T 



(6.8) 



^+1)-m(m+1) 



For the important modes, the right hand side of eq (8) is negligibly small if /x t^v since the 
numerator vanishes at the limits ka and kb when R g and R t approach ± 1 . For the important 
case fx = v, a normalization factor is defined by 

r*kb 

JV.^lim z tx (p)z v (p)dp 

H-^p J ka 

C m 2kh 1 

= L ^( P )Ydp^—, where 5 n = ^ ^ ^ p; (6.9) 



1- 



2kh C n 



It should be remarked at this point that the modes are not strictly orthogonal since the 
right-hand side of eq (8) does not vanish identically although it is small compared to N v . As 
the conductivity of the bounding walls approaches infinity the modes would be completely 
orthogonal. 

Multiplying both sides of eq (1) by z v (p) and then integrating with respect to p from 
ka to kb, leads to the following formula for T) v \ 

^ A^/cosJ >^^- (6J0) 

173 



To actually evaluate D v , it is desirable to let r->r and 0-^0, in which case $(r,d)—>\l/ (r,d) 
where \f/ is the primary influence which is singular at (r ,0). For a vertical electric dipole 
consisting of an infinitesimal element of length ds and carrying a current, /, it is well known that 



TJ S -MR 



where E={r\ J r r 2 — 2rr cos 6) 2 . 



Following the process suggested by Sommerfeld for the determination of the Green's 
function for the perfectly conducting sphere, the integration in eq (10) is carried out in the 
immediate neighborhood of the source. For example, r=r Q (l-\-rj) where — e<C r7<C + € 7 e^^l 
and dr=r dr), z v {kr)^z v {kr^), e~ ikR ^l, while 



Therefore 



Now 



i !S i[(l + ,)- + l- 2 (l + ,)(.-0]- 1 K ^ 9i? . (6.12, 



lim [P,(-cos 0)]-* S111 ^ r log 6 2 



and 



lim (^ d \ =lim/log [e+^e 2 +d 2 ]-\og [- € + V?+#]\^log(? 2 . (6.14) 



It then follows that 



n % z v {kn) Ids . 

^ =z 7rrr~ 1 — °n- (6.15) 

2Jch sin j/7r 4r 



The final form of the function \f/ is thus given by 

iS»a\~ Idsi X^ Z ^ kr ^ 2 ^ kr ) Pv( — COS 0) , ft - A v 

^(r0)~—— 2_j n x 77-7 : o* (6.16) 

2ifo , on=oM... 2, (to) 2„(Z:a) sin j>tt 

where the second-order or W.K.B. representations may be used for radial functions z v (kr), etc. 
As can be seen from eq (4) these can be greatly simplified if z/a < C < C |Cj|, for then 

g,(fcr) _ e ikC n*+R g (C n )e- ikC nZ = 

z v (ka)= 2[R g {C n )$ Jn[) [ ° 

which is the same height-gain function obtained for the flat earth case. For heights even as 
great as 10 km and frequencies less than 20 kc, this is an excellent approximation. Similarly, 

gfgW^o) (6.18) 

where z =r Q —a is the height of the source dipole. 

The radial field component is of most practical interest and, for the moment, attention 
will be confined to it. Since 



and 



E T =U JL |Ysin^) (6.19) 

r sin ddd\ 50/ 

s -hh^[ si " *-^w^> (" +1 > p '<- cos *> =° < 6 - 20) 

174 



it follows that 



*«^S^- W '- W S^^— * } (6 - 21) 



wil h ?+%^ &&#„.. This is t he final solution of the problem being valid for the air space between 
the earth and the ionosphere. 4 

For purposes of computation several simplifications can be made. The asymptotic 
expansion for the Legendre function, given by 

is valid if M>> 1 an d # not near or *"• Since the imaginary part of v(t—6) is also large for 
7T — greater than about 10° or 20°, it follows that 

P > ( " C0S H^ref^ [*(*+§) (-^)-W4} (6.23) 

Furthermore, the source and observer heights are usually sufficiently low that kh C n and 
khiC n <^\ and r {) ^r^a. 

The simplified form of the field can now be written 

^ ^Lsind/oJ (A/X) ^o n n ( j 

where d=ad, the arc length between the source and the observer, X is the free-space wave- 
length, and #„=(1 — Cn)*. E is the field of the source at a distance d on a flat perfectly 
conducting earth. For d/X^>\ , 

E =i (rj/\) Ids (e- i2 * d/ *)/d. (6.25) 

As the radius a of the earth tends to infinity it is immediately evident that the flat earth 
formula given by eq (2.5) is recovered. 

7. Antipodal Effects 

The general form for the field in the space a<V<a + A for a vertical dipole source has 
the form 

fi-SPi £ «. "t±^P,(-cos 6) (7.1) 

where <5 =^, 5 W = 1(715^0), and 

v(p+l)^(v+i) 2 ^kaSl (7.2) 

Now as mentioned above when 6 is not near or x, the Legendre function may be replaced 
by the first term of its asymptotic expansion. This result quoted above is valid if 

rT >(7r-^) and,^»0. (7.3) 

In this region, the modes are simply proportional to 

77—71 cos \kaS n (w-d) -j 1 (7.4) 

(sm e) 2 L 4 j 



4 From an analysis by Pekeris; Phys. Rev. 70, 518 (1040) it may be shown that 

- fit Jlb<» (A'6»- -^(^) H 2 (2) (fcS„p) 



™'*«o0-£(j) 



plus terms in (p/a) 4 , (p/a) 6 , etc., where v-\-\=kS„p. Inserting this in eq (21) leads directly back to eq (3.21) when the curvature correction terms 
are neglected. 

175 



which apart from a constant factor can be identified as the linear combination of two peripheral 
waves of the form 

__i . e ~ikaS n e 

(sintf)* 
and 

1 e -ikaS n (2T-d) e i*l2 

(sin 0)* 
where 0<V. 

These waves are traveling in opposing directions along the two respective great circle 
paths ad and a(2w—6) from the source to the observer. It is noticed that there is a ir/2 phase 
advance which the wave traveling on the long great circle path picks up as it goes through 
the pole 6=w. The linear combination of these two traveling waves is to form a standing 
wave pattern whose distance A m between minimums is approximately given by 

kA m ReS n =7r or A w =A/(2 Re S n ) 
subject to 

-Im S n <Re S n . 

As one approaches the pole d=w, the first term in the asymptotic expansion for the 
Legendre function is inadequate. A more general form is the asymptotic series [27] 

p 2_ jrvfi). m) if cos[Q+^>-*>-(^>] 

" { C Pj= 7r^I>+3/2) Y- (H-3/2),/! (2sm0) J+ « v °> 

wliere 

(a),= a(a+l) (a+2) . . . (o+l-l), 

for example, a =l, ai = a, a 2 = a(a-\-l), <* 3 = a(a+l) (a + 2), etc. 

Since \v\~>l the factorial functions may be replaced by the first two terms of their asymptotic 
expansions; this leads to 

r(H-i) 



K'-i+°(?)> 



T(v+3/2) v \ 

The preceding asymptotic expansion for P v (— cos 6) is not usable at and in the vicinity of 
the pole B— w. In this region a suitable representation is given by [27] 

P,(- cos 0) =Jo(u) + sin 2 (^) [4r~ J2 (") +5 ^ (i)]+0 (sin* (^) ) (7.7) 

where 77= (2y+l) sin [Or— 0)/2], J m (rj), for m=0, 1, 2, and 3, is the Bessel function of first type 
of argument 7? and order m. When w— 6 is small the first term is usually sufficient and further- 
more 

V^(v+i) (ir-d)^kaS n (T-d). 

Thus for mode n, the field in the neighborhood of the pole is proportional to the Bessel function 

Jo[kaS n (T—0)]. 

It is then not surprising to see that the first term of the asymptotic expansion of J is the same 
as that of P v (— cos 0). 

8. Resonator Type Oscillations Between Earth and the Ionosphere 

At extremely low frequencies (elf) , where the wavelength is large compared to the height 
of the ionospheric reflecting layer, the electric field is essentially radial and only one waveguide 

176 



type mode is significant. The field is thus expressed by the first term of the mode series which 
reads 

4kha 2 sin kit 
where v-\-^^kaS Q and 

s^-miw+w) (8 - 2) 

in terms of the relative refractive indices N t and N g of the homogeneous ionosphere and the 
homogeneous ground, respectively. Now at elf |iVg|>>|JVj| and furthermore, 

Thus 

7*3/2 



2 (a^h 

Now as mentioned in section 7, the factor P„(— cos 0) may be replaced by an asymptotic expan- 
sion if kad or kaiir—d) is somewhat greater than unity. The field in this case may be regarded 
as two azimu thai-type traveling waves. Furthermore at the pole (6 near t) where the second 
of these restrictions is violated, it is possible to use an equivalent representation which correctly 
accounts for the axial focusing. An alternate viewpoint which is suitable at elf is to consider 
the field as a superposition of cavity-resonator type modes. It is expected that such a repre- 
sentation would be very good when the circumference of the earth is becoming comparable to 
the wavelength. A suggestion of this kind was apparently first put forth by Schumann [28]. 
The starting point is the expansion formula 

P -4^=- 1 - ± P„(z) . J*+\ _ (8.3) 

sin vw t £=q n(n,-\-l) — v(v+\) 

where the summation is over integral values of n. This result follows directly from a formula 
given by Magnus and Oberhettinger [27] (p. 57) which is valid for vt*0, ±1, ±2, . . . , and 
O^0<7r. The electric field, for A/a<<l, is thus written 

/^+l) ± 2 "+l ( 8 .4) 

where #=cos 9. The early terms of the series are then proportional to 

Po(*) = l, 

P 1 (x)=cos 6, 

P 2 (£)=4(3cos 2 0-1), (8.5) 

and so on. The configuration of the electric field in the first three cavity modes is depicted in 
figure 6. 

Retaining just the first term it is seen that 

K=E r ] =-^~, I (8.6) 

which is independent of 0. Clearly this corresponds to a concentric spherical capacitor ener- 
gized by a current Ids/h resulting in a constant voltage hE° r between the plates. On rewriting 
eq (6) in the form 

hE ° JJ £r (87) 



532053—60- 



177 






Figure 6. Depicting electric field lines in the first three cavity 
resonator modes. 



it is seen that 



a= 



47ra 2 e 



which can be identified as the capacity between the spherical surfaces whose areas are both 
47ra 2 within the approximation h/a<C<Cl. 

It has often been suggested that the omnipresent constant voltage gradient in the atmos- 
phere results from the accumulated action of lightning strokes which impart a charge to this 
earth-ionosphere condenser. For example, when a current surge flows say for 10~ 3 sec with an 
average of 10 3 amp with an average column height of 3 km (i.e., ds ^2X10 3 , then for h^70 km 
it readily follows that 

^1.3v or£'°^2X10- 5 v/m. 

Presumably, many such charges are required to build the field up to its observed value. 

Of somewhat more interest are the cavity-resonator oscillations which may be excited. 
Using the notation of the operational calculus iu is formally replaced by p then 



v(y-\-l) ^ —p 2 —p^a 



(8.8) 



where a=1/[h(<Tifi) l/z \. The source dipole moment Ids is in general a function of time. For 
purposes of illustration, consider 

Ids=(Ids) u(t) (8.9) 



where u(t) is the unit step function at £=0. The Laplace transform of the source moment is 
thus given by 

(Ids) 



/» 00 

Idse~ vt dt= 
Jo V 



The Laplace transform of the field is given by 



w . v (Id8) , iN^o / \ 2n+l 

n °e n=o o)i J r p ZJ r ap 2 

178 



(8.10) 



(8.11) 



where c/ n =(a/c) 2 n(n-\-l). The actual time response of the electric field is denoted e r (t) and is 
zero for /<0. It is related to E r (p) by 

E r (p)= ( e r (t)e- pt dt. (8.12) 

The inversion of this integral equation is a standard problem in operational calculus and has 
been carried out explicitly by Schumann [27] for a transform which has the form of eq (11). 
in the present discussion a much simpler approach is used which is justified when the damping 
is small. It should be noted that ap 1 has already been assumed small compared to p, thus a 
perturbation method is in order. 

When a = corresponding to no dissipation (i.e., perfectly conducting" boundaries) 

The poles in \\w j> plane are thus at p= ±i«„. The inversion to the time domain ^ives 

e^(t)=^^P n (x)(2n+l) cos a>„t (8.14) 

which may be verified by noting that the above expressions for E r (n) (p) and e r in) (t) satisfy eq 

(12). 

A step-function dipole source thus excites the static held (i.e., co = ()) and the cavity-reson- 
ator modes (n=lj 2, 3, . . .). For a =6,400 km 

co,/27r=10.6 cps 

a> 2 /27r=18.3 cps 

co 3 /2tt=25.9 cps. 

To account for finite conductivity it is necessary to solve the equation 

p 2 +pV' 2 a+a>l=0 (8.15) 

which gives the poles for the function E r (p) m ^ ne ( ' asr when a^O. Remembering that ap* 
<C<C:P, it readily follows that 

p^iu n ^+^e iZ *^M n -Sl n (8.10) 

where J n = u n ( 1— r^ — ^ J is the resonant frequency and 

o - <"#* 

^n 23/2 

is the damping coefficient. It then easily follows that cos oo n t is to be replaced by 

e -n n t ( , os ^j 

To this approximation, the effect of finite conductivity is to exponentially damp the oscillations 
with time. For a=6,400 km, h ^100 km, o-^lO" 4 mhos/m, the time constant is given by 



tin ^n(n+l) 
winch is rather interesting. 

179 



sec (ti= 1,2,3, . . . ) (8.17) 



- r+j , 



The total field is thus given by 

« f (^^2P n W(2fl+l)r a .' coso'J (8.18) 

which is valid for a 2 £>> 1 or £>> l/(/i 2 <r^u). 

9. Excitation by Horizontal Dipoles for the Curved Earth 

The formulation of the theory for a horizontal dipole is similar to that for a vertical dipole. 
The complexity of the equations is greater, however, because of the nonsymmetry of the prob- 
lem. Schumann [6] uses this approach in his analysis but his results are not complete as dis- 
cussed below. The deficiency arises when the eigenfunction series is matched to the source 
singularity. In the case of the vertical dipole as outlined in the previous sections, this process 
is relatively straightforward but in the case of the horizontal dipole there is coupling between 
TE and TM modes which apparently is not accounted for using this technique. An alternative 
is to set up the problem in terms of a harmonic series representation wherein the summation is 
over integral values of n, the index of the spherical wave functions. This series is poorly 
convergent, however, and the Watson technique must be used to transform it to a series of resi- 
dues at the complex poles v. Such a procedure was used by Wait [29] for a horizontal dipole 
over an earth with a homogeneous atmosphere. It would not be difficult to generalize these 
results to include the influence of the ionospheric reflecting layer. In the present work, how- 
ever, it seems more instructive to use a different method which makes use of the reciprocity 
theorem and the results for vertical electric and vertical magnetic dipoles. 

For the first part of the problem a vertical magnetic dipole of moment Kds is considered. 
It is located at r=r on the polar axis. Due again to the intrinsic symmetry of the problem the 
fields can be obtained from a single scalar function \j/ n as follows : 



rj i 1 d / - a W\ 



H e =--^(r+ h ) (9.1) 

and H <f> =E r =He = 0. Such fields are purely of the TE type whereas they were of the TM type 
for a vertical electric dipole source. 

The solution precedes in the same manner as for the vertical electric dipole. Now, how- 
ever, the boundary conditions are that (l/r?r)d(r^)/dr and k^/ h are continuous at the concentric 
spherical interfaces. In a form suitable for application to vlf propagation, the final result is 
given by 5 

^=J^ £ 8 ^[ Zo) l^ z) P- y (-cosfl) (9.2) 

2khr m =i,2~3, . .. sin*>7r 



where 



*«~ ,Jn^n ' (9-3) 



I 



sin 2khC,„ 



2khC m 
2/4(z) = [i^«7J]-« exp [ikC m z]+[Ri(CJ)i exp [-iW m z] (9.4) 



5 To conform with standard waveguide practice, the TE mode of lowest attenuation is denoted w = l. 

180 



and similarly for /*(2 ). The modal equation has the form 

R h g (C)R h i(C') exp /-i2A fT(7 2 +— S 2 1 dz\=e~ 2 * im (9.5) 



where 



and 






The electric field component E$ is thus given by 

E ^Kds _ 8h f h m (zo)fl(z) dP-(-cos 6) 
2Ar to m sin vt dd 

Now when the source is a small loop of area da earning an average circulating current / it 
follows that 

Kds=ifxo)Ida. (9.7) 

Furthermore, if the receiving antenna is a horizontal electric antenna of effective length dl, 
the voltage at the terminals is given by 

v=E*dl sin <f> (9.8) 

where <j> is the angle subtended by the receiving antenna and the arc joining the two antennas 
(see fig. 7). Thus the mutual impedance Z m between the source loop of area da and 



/ 

/ 

source (P ,an view ^ /^ Figure 7. Source vertical magnetic and electric dipoles and 
dipoles ^| \S \<p receiving horizontal electric dipole for mutual impedance 
®- * — y£- calculation. 

I a0 -J 



the horizontal receiving antenna of length dl is 

y _v _ ip<»> da dl sin <l > ^ , Q QX 

^~I— -2htT " v " ( } 

where the summand is the same as in eq (6). 

The mutual impedance Z e between a vertical electric dipole source at 0=0, r=r , and the 
horizontal receiving antenna is also required. This may be obtained from the scalar function ^ 
previously obtained. In particular 

"--I'm? w^'dsi* <9 ' 10 > 

181 



where 



-R^ exp [-*£ (Cl+f Slf] <**}• (9-12) 

When z/a«\C 2 n \ 

2g n {z)5±C n [R-\ n e ikC »°—R u g i e- i * c n*]. (9.13) 

Furthermore if |&<7„2| <C<0 which is the usual case 

g n (z)^A„f n (z) (9.14) 

where 






w, 



The mutual impedance Z e between the vertical electric dipole and the horizontal receiving 
antenna dl is thus given by 

z „ v ds dl cos t ^ (915) 

2/ir n 

where the summand is the same as in eq (11). 

It is now a simple matter to write down the field expressions when the source is a hori- 
zontal electric antenna carrying a current / of length dl. The antenna or dipole now is con- 
sidered to be located at r=r and 0=0 and oriented in the direction 0=0. The vertical magnetic 
field at (r, 6, <j>) is obtained from the relation 

ifiuH r da=IZ m (9.16) 

which relates the total magnetic flux in a small loop of area da at (r, 6, </>) and the vertical 
magnetic field at the same point. Using eq (9.) it is seen that 

tt Idl^. h f^(Zo)fi,(z)bP7 (-cos e) . , Q17 , 

In a similar fashion, the vertical electric field at (r, 6, <£) is obtained from the relation 

E r ds=IZ e (9.18) 

which relates the voltage in the small vertical antenna of length ds at (r, 6, </>) and the vertical 
electric field at the same point. Thus 

^fs^^-Hos S) cos,. (9.19) 

Ihv n sm vir 06 

The other field components can be found from the above expressions for E r and H r . Quite 
generally the field components in spherical coordinates (see fig. 8) can be written in terms of a 
set of purely TE and TM modes derivable from scalar functions U and V. Thus 

*-P^?W] (9.20) 

Ee== l -V m - f/W (9.21) 

r d0dr v ' r sin d<£ v 

182 




\ e *( fi ch) Figure 8. Spherical coordinate system for horizontal electric 
°g £ g y. /\ U,o>.9/ dipole between concentric spherical interfaces. 






r sin 5 d</>dr 

Since U and V satisfy the equations 

(V 2 +F)£=0 

in the space a<V<a+A they must be made up of solutions of the form 

hl 1] (kr) cos #0 

P?(cos0) 

A< 2) (&r) sin g<£ 

where q is an integer. Since the field for E r and H T has already been prescribed, q= 
some consideration it is seen that 



(9.22) 
(9.23) 
(9.24) 
(9.25) 

(9.26) 



and 



Further, on noting 



it is seen that 



and 



tt ^-^ a r / \ &Pv (— cos 6) 

U=Y^A n j n {2) ^ } - COS* 



i7 x^ t> -cn ( \ <*P* ( — cos (9) . , 
^=]E -B w /» (z) ^ sin 0- 



[* 2 +^-^]w- 2) (^)]=o 



c ^•'(''+1) ,, , , ^ 5/ J ,(-cos ») , 

#r = Z! f -4»/»(s) "^ L COS <*> 



tt ^ v{v+\) t> A /„\ d7 J ;(-cos 6>) . 

«r=Zj r ii m j h n {z) - —r- sin <£• 

„i r on 



(9.27) 
1. With 

(9.28) 

(9.29) 

(9.30) 

(9.31) 

(9.32) 



183 



On comparing eqs (30) and (31) with (17) and (19), it follows that 



and 



A-T^htin^ (9.33) 

71 2hv(v+l) sin vw v ' 



T> ___Z____ °mJ m\Zo) /q oa\ 

nm ~2hv(T+l) sin?7r " K " M) 



When |*>|>>1 and 6 is not near or it the following asymptotic expansion is valid: 

d P„(-cos 0) g* 2e ~ %T '\ (ffsi /2 e- ikadS « (9.35) 

(sin 6)* W 



sin z>7r dQ 
where use has been made of the relation 



ikaS n . 



The height function g(z ) occurring in the expression for E r can be simplified at low heights. 
For example, if |&C W 2 | < ^ < which is the usual case 

^^wX'-mf' (9 - 36) 

Thus the vertical electric field of a horizontal dipole is well approximated by 

K^Eo ^^P^rT ^vtkv e« w ^- (f/4, ]][; b n S l J 2 e- i( * s nW (9.37) 

N K [sm d/aj (h/\) £=i 

where 

E =i(r)/X)Ids(e- i2 * d ^)/d. (9.38) 

It is of interest to compare this with eq (6.24) for the vertical electric field of a vertical dipole 
with the same moment. It is seen for a given mode 



Ej n) (for horizontal dipole) _ cos <f> / _SV\i 
E™ (for vertical dipole) ~~ S n N g \ N*J 



(9.39) 



Since S n ^l, it is seen that the ratio does not depend critically on mode number n, thus 

E r (for horizontal dipole) cos <j> / 1\1 , Q A() , 

E r (for vertical dipole) ~ N g \ N\) ' [ } 

In most cases |2V & |>>1 so the ratio is of the order of l/N which is small. In particular, at 
very low frequencies 

cos <i> 



N g 



*-(¥f eiT/ * cos * (9,41) 



which indicates that the ratio varies as the square root of the frequency. This is in disagree- 
ment with the results of Schumann who finds that dependence is the inverse first power of 
frequency. In the direction </>=0, of course, the ratio derived here turns out to be nothing 
more than the "wave-tilt" for a vertically polarized plane wave at grazing incidence on a flat 
earth. Thus, Schumann's results [7] for the horizontal dipole would seem to be in error. 

The horizontal dipole also, of course, radiates horizontal polarization. The simplified 
expression for H T can be written by employing the single term asymptotic representation 
described above. Thus 

H ^ E sm<j>{dl\^ r dja -|* e<t( , rt/x) _ ( , /4)1 £ sie -iws m S%fl{z a )ft{z). (9.42) 

V WIN L sin «/ ffl J m=l 

184 



10. Influence of the Earth's Magnetic Field 

In the preceding analysis the earth's magnetic field has tacitly been neglected. To 
indicate its effect the reflection coefficient for sharply bounded ionosphere with the magnetic 
field included shall be discussed. The derivation of the general formulas is due to Budden [30] 
but for highly oblique incidence great simplifications to his results can be made. 

The starting point is the magneto-ionic formula of Appleton and Hartree for the complex 
refractive index /* for a homogeneous ionized medium with superimposed magnetic field. 
In the region from 70 to 90 km in the ionosphere where very low frequencies are reflected, it 
is often permissible to employ the quasi-longitudinal approximation of Booker. It is now implied 
that the waves after they are transmitted into the ionosphere are steeply refracted toward the 
vertical. Essentially this means that the refractive index does not depend to any great extent 
on the direction of propagation for temperate and polar latitudes so that 

M 2 ^l-i(co» exp (±ir) (10.1) 

where 

tan t=oo l /v 
and 

In the above 

u>l=Ne 2 /em, 

N= number of electrons per meter 3 , 
e and m= charge and mass of electrons, 
e=8.854Xl0- 12 , 
y= collision frequency, 
co L = (4ttX 10" 7 )^/m, and 

H— effective strength of the earth's magnetic field, (i.e., the longi- 
tudinal component for propagation in the ionosphere). 

It is now desirable to consider four reflection coefficients \\R\\, \\E±, ±R\ \ , and ±R± to indicate 
the complex ratio of a specified electric field in the wave after reflection to a specified electric 
field in the wave before reflection. The first subscript denotes whether the electric field 
in the incident wave is parallel (||) or perpendicular (J_) to the plane of incidence and the 
second subscript refers in the same way to the reflected wave. A cartesian coordinate system 
(x, y, z) is now taken with z measured vertically upwards. The incident wave has its normal 
in the xz plane inclined at an angle 6 to the z axis. The components of the electric field are 
E\\ in the xz plane and E± perpendicular to this plane (i.e., in the direction of increasing y). 
When the -f sign is taken in eq (1), the refractive index is denoted /x corresponding to the 
ordinary wave, and when the — sign is taken the refractive index is denoted id e corresponding 
to the extraordinary wave. With this convention, it can be shown that 

E ±0 /E { \ =-i and E ±e /E lle =i (10.2) 

in the northern hemisphere. 

The incident wave is now characterized by a factor exp [— ik(x sin 6-^-z cos B)x\ and the 
reflected wave, therefore, contains a factor exp [—ik(x sin 6—z cos0)#]. Furthermore, the 
transmitted waves have factors exp [—ik(x sin o +2 cos o )/xo#] and exp [—ik (x sin e +2 cos d e )n e x]. 
The reflection coefficients are now obtained by matching tangential field components at the 
air-ionosphere interface. The results, expressed in a form suitable for computation are 
listed below. 

\\R\\^[^o+^)(C 2 -C C e ) + (^ e -l)(C +C e )C]/D (10.3) 

\\R ± =2iC(fji () C -^C e )/D (10.4) 

±R u =2iC(njC § -p.Co)ID (10.5) 

±R±=l(^+fJie)(C 2 -C C e )-(^ e -l){Co+C e )C]/D (10.6) 

185 



where 
and 



D=(n +» e )(C 2 +C C e ) + (M e +l)(Co+C e )C 

C=cos 0,Co=COS 6 Q ,C e = COS 6 e . 



(10.7) 



Numerical values based on these formulas are available. 

Now for highly oblique incidence the value of \C\ is small. Thus for |(7 2 |<C1 



.finS- 



(Mo+Mg)CoPg 
1 , (MoMe+l)(^o+fi) c 



Furthermore if |jLt MeC y2 | < ^l, 



ii xi ii = 



a _[, 



(Mo+Me)CoC e 



2/XoMg ^O+^e 



^-exp(-2ft,C) 



°] 



(10.8) 



(10.9) 



where 

To the same approximation 

where 
Also 
and 



flr 



M0 + Me Co^ 



MoM<- 



L Mo+Me <-,oC e J 



&.= 



1 c +c e 



H0 + /JL e C Q C e 



p 2iC fioCo—fj^ 

^0< e M0 + ^e 



■iA* 






Moi-Me 



(10.10) 

(10.11) 
(10.12) 
(10.13) 
(10.14) 



It is immediately evident from the above that as tends to x/2 (i.e., grazing incidence), 
the reflection coefficients ||i?n and ±Rx are both approaching — 1 whereas the conversion co- 
efficients | f jB_l and ±R\\ are both approaching zero. In this sense a sharply bounded ionosphere 
behaves as an equivalent isotropic medium for highly oblique incidence. 

Some further simplifications are possible when the ionosphere is effectively a good con- 
ductor. For example, if 



Consequently, 



and 



|co r /co|^>l 
jug=l — i(u r /oo)e iT ^ — i(co r /o))e iT , 
f4 = l — i(u r /u>)e~ iT ^:—i(a) r loo)e~ iT . 

(l + i)(u/u r )i cos (r/2)(C 2 -l)±2^[l-^/co r ] C 
!lPr* (l + i)(coK)* cos(r/2)(C 2 +l)+2^[l + W^] C 



B..S 



,#. ^ 



-2(a;/co r )^(l + ^)Csin(T/2) 



!i ft (l + i)(coK)icos(r/2)(C 2 +l)+2i[l + ^/co r ]C 

186 



(10.15) 
(10.16) 

(10.17) 
(10.18) 



These may be further approximated, for cos 0«1, by 

1 _ I| ^^ 2 (^ C - OS ^, ± A' ± +1^2( / ^y^^ (10.19) 

\o) r / COS 6 \u) r J COS (r/2) 

provided the right-hand sides of these equations are small compared to one. To the same 
approximation 

!i ^- L «-2(i«/^)* sin (r/2). (10.20) 

±#11 

The quasi-longitudinal approximation used above is only valid when 

iM-'-yi 



w S«i 



4co 2 co| 

where co L and co r are tlie longitudinal and transverse components of the (angular) gyro fre- 
quency. Clearly, this condition is violated when the transverse component of the earth's 
magnetic field is large such as for propagation around the magnetic equator. 

The case of a purely horizontal and transverse field has been considered by N. F. Barber and 
D. D. Crombie (to be published in Journal of Atmospheric and Terrestrial Physics). Their 
results, applicable to a sharply bounded ionosphere, may be written in the following form 

II fl II =£=!>!! #l=0 (10.21) 

where 

A ~ {i+iLy-r 

where L=u/o) r and 7=co r co/a>o. For east-to-west propagation (along the magnetic equator), 
7 is positive, while for west-to-east propagation, y is negative. For highly oblique incidence, 
the above simplifies to 

\^P l} (10.22) 

where 

1 (l + JL)l(iL-L 2 -y*)l-iy 
Al (l+iiy-y 2 

Furthermore if, |/3| 1 C r |«l 

B«ll as -«-*"* (10.23) 

which has the same form as equation (11) 

The exact determination of the reflection coefficients for an y orientation of the earth's 
magnetic field may be carried out using a method outlined by Bremmer [3]. This has been 
done b} r Johler and Walters whose results are to be published in the following issue of this 
journal. 

1 1 . Mode Series for an Anisotropic Ionosphere 

In this section the formalism for the mode theory is developed for a plane earth and an 
anisotropic sharply-bounded ionosphere. The geometry is the same as in section (3) where the 

187 



ionosphere was assumed to be isotropic. In the present case the reflection coefficient [i?*]" is 
regarded as a matrix and written in the form 

™"-[& jST" m 

where the two primes are to indicate that it is a two column matrix. The individual coefficients 
l|i?U, ± R\\, \\R±, and j_R ± discussed earlier, indicate the complex ratio of an electric field com- 
ponent in the wave after reflection to an electric field component in the wave before reflection. 
The first subscript denotes whether the electric field specified in the incident wave is parallel (||) 
or perpendicular (±) to the plane of incidence. The second subscript refers in the same 
way to the electric field in the reflected wave. 

When the ionosphere becomes isotropic corresponding to a zero magnetic field, the reflec- 
tion coefficient in matrix notation becomes simply 

where R t and R* are the complex scalar reflection coefficients for vertically-polarized and 
horizontally-polarized waves, respectively. The corresponding [matrix] reflection for the 
ground is 

i*j"=[? y- (n - 3) 

The case of two successive reflections, the first from the ground and the second from the 
anisotropic ionosphere, is represented by the matrix 

[R g R i ]" = [B i ]"X[R s }"=[^ j£j_ *j ^}" (11.4) 

In the case of the isotropic ionosphere this reduces to 

[W«[V bJjj}" (11 - 5 > 

The arguments employed here are virtually identical to those of Budden who, however, assumes 
a perfectly conducting ground, such that 

# t =+landi?»=— 1. 

In the previous formulation for a vertical electric dipole between the plane ground surface 
and a sharply bounded isotropic ionosphere, the fields could be completely derived from an 
electric Hertz vector which has only a z component. Of course, if the source was not sym- 
metrical it was necessary to introduce an additional component of the electric Hertz vector. 
When the upper boundary is anisotropic, the single component Hertz vector is not adequate 
even for a vertical electric dipole source. This is not surprising since the TM modes are coupled 
to the TE modes by the anisotropic boundary conditions. 

Any electromagnetic field, in such a parallel plate region, can be obtained from a super- 
position of TM and TE modes which are derived from electric and magnetic Hertz vectors. 

188 



These are denoted individually by II [j and II ± or, collectively by the matrix 

[n]"=Rj"J (n.6) 

where the single prime is to indicate that it is a single column matrix. II ± which is a magnetic 
Hertz vector is often referred to as a Fitzgerald vector. Furthermore, the electric and magnetic 
fields can also be written as single column matrices in the manner 

A'-dJ.i3r-[fJ («•») 

where 

2?ll=(P+graddiv)il|i (11.8) 

H\\=Uu curln|] (11.9) 

vH ± = (F+grad div)n ± (1 1.10) 

-^ -> 

rjE ± = — i^ curl n_L. (11.11) 

-» -> 

The intrinsic impedance 77 is introduced in the latter two equations to make II|| and II x of the 
same dimensions. The Hertz vector in matrix form corresponding to the primary excitation is 
then written 



HH 



(11.12) 



where U p has only a z component IT ( ? ) . To match boundary conditions in the case of azi- 

muthal symmetry it is only necessary that the vectors II|| and II ± have a z component. The 
condition of azimiithal symmetry is achieved if the reflection coefficients themselves are inde- 
pendent of the azimiithal coordinate <£. 

Formally the solution has the same form as the isotropic case if the appropriate reflection 
coefficients are now regarded as matrices. For example for the space o^z^h the (matrix) 
Hertz vector 

[l h y^ l MMyC[F((^]^H^(kS P )dC (11.13) 

where 

Ids 



W-flfJ 



with M=- 



4irl eco 
and where 

L lWJ e ikCh {\-[R g R l Y f e- 2ikhC ) K } 

It should be noted that the denominator in the above expression is also a two column matrix. 
Inverting this, following the usual rules for such operations, leads to 

\F{(i)Y' = {e ikCz +[R g Y f e- ikCz ){\l^[NY (11.15) 

for £ o =0, where 



A= 



««-|fl| R t 


j.B| B\ 


0« x R g 


e 2iC k H__ LR± flA 


189 





(11.16) 



and 

The corresponding residue series are thus given by 

n J =- iM ? I5Akv7, Ws "> [el'J (1118) 

with 

Gy=f p (z)(e 2iC * kh -R h g ±R±) (ll.W) 

and 

G ±P =ift(z)R*\\R±. (11-20) 

The summation is over the poles of the integrand [^(C)]"- Clearly this corresponds to the 
roots of the equation, A = 0, which are designated C=C P . It is understood that all quantities 
in the summand of eq (18) are to be evaluated at C=C P . The "height-factors"/^) a,ndfl(z) 
have the usual form, that is 

2j p (z) = (R g )-ie ikC » z +(R g )ie- ikC * z (11.21) 

and 

2f*(z) = (Rl)-le aG *+(Ri)le- tw *. (11.22) 

The above results reduce to those of Budden when the ground is perfectly conducting. 

The modes excited in the waveguide can be logically grouped into two sets. The first 
has a TM (transverse magnetic) character and the second has a TE (transverse electric) 
character. To obtain the attenuation and the phase constants of these individual modes, 
it is adequate to consider the anisotropy as sort of a perturbation to the corresponding TE and 
TM modes for the isotropic case. 6 

To simplify the discussion the ground is considered to be perfectly conducting. That is, 
R g =l and R h g = — \. The modal equation now becomes 

(^^-Il/? 1 ,)(^ 2 ^+ X 7? i .) + , ] 7? J . X 7?|,=0. (11.23) 

When mode coupling is disregarded this breaks into two equations 

{l R l]e -i2Wh =1 = e ~i2, n (11.24a) 

± R ±e - i2kCh =-l=-e- i2Tm (11.24b) 

where n and m take integral values. As mentioned in the previous section the reflection 
coefficients for highly oblique incidence may be approximated by 

iRlOt-e^f (11.25) 

and 

± R ± ^-e~^ ± C (11.26) 

where to a first order, /S|| and /J x are independent of C. It thus follows that the first approxi- 
mation (indicated hx a superscript (1)) for the solutions of the modal equation are 



win— v) 



C,«C«»=J£^ (n=l, 2, . . .) (11.27) 

for the TM modes, and 

irm 
kh—iPt 



C^C^j^T (™ = 1, 2, 3, . . .) (11.28) 



for the TE modes. These have exactly the same form as when the ionosphere is assumed 
to be isotropic. The difference lies in the value of the coefficients /3\\ and fi ± which are functions 
of the earth's magnetic field. 

6 It should be noted that the negative order modes in the case of an anisotropic ionosphere are not the same as tho positive order modes. 
That is, C-i^-Co, C- 2 ^-Ci, etc. 

190 



A second approximation to the mode equations is obtained in the following way. The 
modal equation is rewritten in the two equivalent forms 

l-\\R { \e- i2kCh =2b{C) (11.29) 

l + ±R±e- i2kch =2y(C) (11.30) 

where 

2«(g)— ',^+% JL (H.31) 

and 

I? 1> H „-V2kCh 

2 ^)~ «««*- n tf D < 1L82 > 

It is to be expected that 8(C) is a small quantity for the TM type modes and 7(C) is a small 
quantity for the TE type modes. The second approximations then are obtained by replacing 
8(C) by 8(C^) for the TM set, and replacing y(C) by y(C%) for the TE set. Solving eqs (29) 
and (30) with these substitutions leads readily to 



for the TM type modes, and 



c „ c ^(n-h)-i8(CV) ai33) 



p== Lm kh-ip ± [n - n) 



for the TE type modes. 

These should be adequate solutions since \8(C n a) )\ and \y(C m {1) )\ are small compared to unity 
for the important modes. In fact for most cases of practical interest, the first-order approxi- 
mations should suffice. 

To provide some idea of the character of the TM and the TE type modes excited by a 
vertical dipole source, the ratio of the tangential magnetic field in the two principal planes is 
considered. For the pth mode, this ratio is given by 

(tM^")/(-«»^') 

for kp»l 



ikll\\ 

C,\\Bi.B t 

[e 2iCkn -R h gS _Rx\ 



kC p dz JA *> 



(11.36) 



where it is understood that the reflection coefficients are to be evaluated at C=C V . The pre- 
ceding expression reduces to 






which was given originally by Budden [11]. In general this ratio is small except near the TE 
resonance wherein the denominator becomes very small. <For the important TM modes which 
are of low order, both C v and \\R ± are small compared to unity and thus the magnetic field 
H p in the direction of propagation lias a relatively small magnitude. 

12. Higher Approximations to the Curved Earth Theory 

In the previous sections the mode series for a concentric spherical earth-air-ionosphere 
system was developed. In order to simplify the discussion and lead to results suitable for 

191 



immediate use, rather crude approximations were introduced. In this section the problem is 
reformulated in a more rigorous fashion and higher order approximations for the various spher- 
ical wave functions are introduced. This analysis is really an extension of the work of Watson, 
Rydbeck, and Bremmer. The final results indicate the range of validity of the lower order 
approximations used in the earlier sections. The formulas are in a form which is suitable for 
numerical computation. 

The earth is represented by a homogeneous sphere of radius a and is surrounded by a 
concentric homogeneous sharply-bounded ionosphere of radius c. The source is a vertical 
electric dipole of strength I ds and is located at r=b. The electrical constants of the air space 
are denoted e and n and subscripts g and i are added to these when reference is made to the 
ground and the ionosphere, respectively (see fig. 5). 

The fields can be expressed in terms of a Hertz vector which has only a radial component 
U, and thus, for the region a<r<a+/&. 

r orod 
E t =Q H,= -i^ (12.1) 

where k= (€ju) 1/2 co. 7 A subscript g and i are also added to the field quantities when reference 
is made to the regions r<^a and r>c, respectively. Furthermore, kg=(e g tXg) 1/2 03 and k f = 
(€ijii;) 1/2 co are the respective wave numbers for these two regions. 
The Hertz functions satisfy the inhomogeneous wave equation 

W+WU=C H :- b \ d ® (12.2) 

2wr sin 

for a<r<a+A, where the 5's are unit impulse functions. The factor 2wr 2 sin 6 is the Jacobian 
of the transformation from rectangular to spherical coordinates. The constant C is to be 
chosen so that U has the proper singularity at the dipole, that is 

fi -ikR 

- bU-> A . p I ds for R-*0 

where R=[r 2 -\-b 2 — 2br cos 0] 1/2 , and therefore 0= (i/we)/ ds. 

The field in the region a<^r<^a-\-h is now written as the sum of the two parts C7 e +L r s , 
where U e has the proper dipole singularity at R—0, and U s is finite at the point. As U s is a 
solution of the homogeneous wave equation, it can be written in the form 

U s =f~ S (2<Z+ 1) [A q h? (kr) + B t j, (kr)]P 9 (cos 0) (12.3) 

47T g = Q 

where j g (kr) and h q m {kr) are spherical Hankel functions of the first and fourth kind, respec- 
tively, and P ? (cos 6) is the Legendre function. The summation is over positive integral values 
of q. The corresponding expression for U e is given by 

ihC °° (j Q (kr)h {2) (kb)) forr<6 

U e =-^-^(2q+l)P q (cosd)\ jQK ^ (12.4) 

^ *=o I hf{kr)j q (kb)) forr>6. 



7 The function U=—it]\f/ in terms of scalar function ^ used previously for the potential. 

192 



Since there are no singularities other than the source dipole, the Hertz functions U g and Ui are 
solutions of the homogeneous wave equations 

(V 2 +k*)(rU g )=0 for O^r^a (12.5) 

and 

(V 2 +k f i)(rU i )=0 for r^c, (12.6) 

Noting that U g is to he finite at r=0 the solution must be of the form 

U g =-f-i: (2g+l)Z',(cos 0)aj 8 (^r) (12.7) 

where a ? is a coefficient which is independent of r and 0. Furthermore, since £7* is to give rise 
to an outgoing wave at r=oo, the solution is of the form 

Ui=~T> (2 2 +l)P ff (cos e)b q h^(k t r) (12.8) 

where 6 ? is a coefficient. 

The four unknown coefficients A q , B q , a q , and b q , can be found from the boundary condi- 
tions at r=a and c. These require the continuity of the tangential field components. In 
order to facilitate the solution and to readily permit later generalizations, the (four) boundary 
conditions as stated above can be replaced by two impedance type boundary conditions. For 
the qth terms of the series these read 

E^ = -Z ( ^H { ^ at r=a (12.9) 

and 

fi}« ) =ZJ f) HJ f) at r=c (12.10) 

where 

£<<?>=_ r_ 

* ieu rj q (k g r) 



and 

1 "dr 



WW)] 



ZJ«>=- 



iew rh q 2) (kir) 
Replacing /tr by #, eqs (9) and (10) may be rewritten 

1 d 



and 



x (xU)=i(ZPh)U iorx=ka (12.11) 



Il(i/) = -i(Z<«>/ij)[/ {oix=kc. (12.12) 



Applying these to eq (3) enable A„ and B q to be obtained explicitly in terms of known quantities. 
Using these results leads readily to the following exact solution for a^r^b. 

U s =-^ S {2q+\)h^(kb)h^ (kr)P t (cos 0) ff (12.13) 

193 



where 



F In p(,) ^(fat) ^(fe-nr ff( ., Ay>(fe) ft}"w i , 191 ,, 

n -i _ pit) p(«) ki™l ?iJM fioui 



ln'[kaK l) {ka)]-iZ^lr, 
~ln'[kah™(ka)]-iZf/v 






fo'[W(fcc)]+iZj"/n 
1 ' Jn'[icA<»(ifcc)]+iZ}"/ij' 

The symbol Zn' denotes logarithmic differentiation, for example 

d 



(12.17) 



W(*)] 
ln'[kah^(ka)} 






aui^- dx _ I ( 12 ' 18 ) 



The above result, although rigorous, is not of practical value for vlf propagation calcula- 
tions because of poor convergence of the series solutions. In fact, something of the order of 
2 ka terms are required to achieve 5 percent accuracy. At 15 kc, for example, 2 ka ^2, 000 
which is a rather large number. An important observation, however, is that terms of order q 
beyond 2 ka contribute little to the series. Thus the spherical Bessel functions j ff (k g a) may be 
replaced by the Debye or second-order approximation since \k g a\^>^>q in the important range of 
q so long as |A: g |>^>fc (i.e., well conducting ground). Thus 

ln'[k g aj Q (k g a)]^i [l-fTf • (12.19) 

Similarly, for |Z?i|>>& 

ln'[k<ah?(k<i)]**-i [i-fTf* ( 12 - 2 °) 

Since the total field is of the form 

U=±(2q+l)f(q)P Q (cose) (12.21) 

q = 

it can be rewritten as a contour integral over q where the integrand has poles when q takes 
integrand values. Such a representation is 

f/ = i Jc 1+ c 2 cJsV 2 "^-* [C0S(T - e)] (12 ' 22) 

where the contour Ci + C 2 encloses the real axis as illustrated in figure 9. Noting that the poles 
of the integrand are located at q= 1/2, 3/2, 5/2 . . . etc., it can be readily verified by the theorem 
of residues that this integral is equivalent to eq (21) . Now, subject to the validity of the second- 
order approximations, for the wave functions of order k s a and k { a mentioned above, the function 
/(#-/£) is an even function of q. This means that the part of the contour C\ just above the 
positive real axis can be replaced by C{ which is located just below the negative real axis (see 
fig. 9). The contour C{-\-C 2 is now entirely equivalent to L, a straight line running along just 
below the real axis. Replacing q~-y 2 by v the contour representation for U takes the form 

U=-i f ^±D f( v )P, [cos (tt-6)] dv. (12.23) 

J^sm v-k 

194 



It is to be noted that this manipulation of the contours is only strictly justified when/(g-}£) is 
an even function of q. This is well justified when \k g \ 2 and \kt\ 2 are both>>F. 



_^v 



Figure 9. The contours in the complex (j plane. 



The next step in the analysis is to close L by an infinite semicircle in the negative half- 
plane. The contribution from this part of the contour vanishes as the radius of the semicircle 
approaches infinity because of the exponentially decreasing character of the integrand. The 
value of the integral for [/along the contour L is now equal to — 2wi times the sum of the residues 
of the integrand evaluated at the poles of f(v) located in the lower half-plane of v. ft then 
follows that 



U=-ikCJ2^^h ( l\kb)h ( l ) (kr)^I\[vos( 7 r-d)} 
v SID vir D v 



(12.24) 



where Dl n = dDvfov. All quantities in the summand are to be evaluated at the poles of /W 
which are the roots of the equation 

D,=0. 

This equation is precisely the same as the one discussed earlier (i.e., eq (5.5)). At that time 
the relevant spherical wave functions of order ka and kc were simplified by the use of the 
Debye or second order approximation. The Hankel or third order approximation will now be 
employed. It may be written [26] 



where 






and H[%(p) is the Hankel function of order 1/3 of argument p given by 



3 L(»+i) 2 J 



For Re (y-\-%)<C<Jz or for jp|>>l, the above reduces to 

to (i) 

xh™(x)Q*S™(x) 

which is the Debye approximation used in section (5). 

The third order representation for the logarithmic derivative is 

L x J m%(p) 

while the corresponding second order approximation, valid for |p|>>l is simply 

ln'[xhp(x)]c* ±?f 1-^^T- 
195 



(12.25) 

(12.26) 

(12.27) 
(12.28) 

(12.29) 
(12.30) 



For convenience in what follows, it is desirable to introduce two new spherical reflection 
coefficients r g and r t which are connected to R g and R t in the following manner : 



lAa 



ln'[kahl 2) (ka)] 

'ln r [kah^(ka)] 



R e = 



In'llcahPQca)] 



^A t , 



and 



where 



and 



In'lkchPjkc)] 

ln'[kch!, 2) (kc)} 



R t =- 



ln'[kahl 2) (ka)] 

iAi 
' ln'[kcM 2) (kc)\ 



iA t 



1 ln'[kch ( v l) (kc)} 



(12.31) 



(12.32) 



(12.33) 



(12.34) 



These new reflection coefficients may be expressed to a high order of approximation by using 
the Hankel or third-order approximation for the spherical Bessel functions of argument ka 
or kc. Thus 



b-mi 



«-«• 



my 3 ( Pa ) 

HilKpa) 



b-mi 



m%( Pa ) 



(12.35) 



e'6 



and 



i-mr 






A t 



il e 6 



where 

and 

To this same approximation 

and 

When \p a \ and |p c |>>l 



b-mi 

y+h f (ka) 2 
Pa ~ 3 |>+§) 2 J 

v+hV (kc)' 2 ,1 

Pc ~ 3 l( V +hr J 



t i Hl%(Pc) 
' H$l( Pe ) 



(12.36) 



3/2 



3/2 



H[UPa) m)i(Pa) g 
<= e flJHGO ff$(p,) * 



r ~#,, 



'D-CSXT 



(12.37) 
(12.38) 
(12.39) 
(12.40) 

(12.41) 



+A g 



196 



and 

f ,ttff,a l )"'£ (12.42) 

On writing v+ \ =kaS= kcS', these latter forms are readily identified as the Fresnel reflection 
coefficients 



and 



>J{ ^ XVgV 'z-i \tlA " J (12 43) 

* N 2 g (l-S 2 )i+(N 2 g -S 2 )i 



mi-(s'rt-m-mv . (12 44) 

1 l m[i-(s') 2 ]*+[m-(sy]i 



Attention is turned specifically to the determination of the roots of the equation 

Z}„=0. (12.45) 

This may be written 

P P h* (ka)h v (fCC) _ 2irin (^9Aa\ 

hgiXt h?Kka)h^{kcr e (lZM) 

where R g and R t are defined by eq (31) and (32) and n may take integral values. Emptying 
the third-order approximations, this may be rewritten, for Re (v-\-^)<Cka 

fM^EMW f -i2(y c -7 a ) f i2(p c - Pa )_ -i27rn /i 2 47) 



where 



and 



■£.['-W* 



while, for &c>Re (v-\-\)^>ka, 



In the above formulas the (spherical) reflection coefficients r g and r t are defined by eqs (35) 
and (36). When \p a \ and \p c \ >>1 or if Re (i , + i)< < (k, the relevant equation for the modes 
is simply 

r r ie - i2(y c-^= e -^n (12.51) 

where r g and r t are defined by eq (41) and (42) which are the Fresnel form of the reflection 
coefficients. Equation (51) is identical to eq (5.10) which was discussed previously. 

13. Influence of Stratification at the Lower Edge of the Ionosphere 

Attention in previous sections has been largely confined to a sharply bounded homogeneous 
ionosphere. In view of the general uncertainty about the electrical properties of the lower 
edge of the ionosphere, a more elaborate model might hardly seem worthwhile. Furthermore, 
despite the geometrical simplicity of the above models, the computation of the modes is very 

197 



involved in the general case. Despite these disparaging remarks the inhomogeneity of the 
lower ionosphere may be considered in some cases without greatly increasing the complexity. 
Some of these generalizations are discussed here for what they are worth. 

The theoretical treatment given in section (3) for a vertical electric dipole located in the 
air space between a flat ground and a plane interface of a homogeneous ionosphere may be 
easily generalized to a stratified ionosphere. The essential modification is to replace the 
ionosphere reflection coefficient Ri{C) by a more elaborate form which is denoted Ri(C). For 
example, a two layer ionosphere is chosen. The lower edge is at z=h and from there to 
z=h+s, the refractive index (assumed constant and isotropic) is Ni) at this point the refrac- 
tive index (also assumed constant and isotropic) is N 2 and remains at this value thereafter. 
It is not at all difficult to show that, for vertical polarization [31], 

E(C). MC-jNl-SVQ . 



where 



Q^ NUNt-S^+NKNj-S^ tanh [iks(Nf- S 2 )^] , 2) 

Ni(m-S 2 )i+m(Nl-S 2 )i tanh [iks(Nl-S 2 )i] 

Here it may be observed that if \kNis\ <C«0 the reflection coefficient becomes 



n JtlC-(Nl-S^ 133) 

v NlC+(Ni-S 2 ) k > 



whereas if \kNis\ >>1, it becomes 

- n 2 g—(N 2 —s 2 )* 

N 2 C+(Nl-S 2 )* 

These two limiting cases correspond respectively, to the conditions of an electrically thin and 
an electrically thick stratum. In the former case the effective reflection level is at z=h-\-s 
and in the latter case it is at z=h. 

The formula for Rt(C) may easily be generalized to any number of layers. For example, 
in the case of discrete layers or strata, 0<2<7*, corresponds to the air; h<Cz<Ch+Si corresponds 
to a stratum with index iV~i, h+Si<C^ < Ch J rSi+s 2 corresponds to a stratum with index N 2 , and 
so on. (See fig. 10.) With this generalization, Q, in eq (1), is to be replaced by 

_ iVf(^I-^ 2 )^ 2 +iVl(Arf->S 2 )Hanh[fe 1 (^-^)i] 
Nl(m-S 2 )> + m(N 2 2 -S 2 )*Q 2 tanh [iks^Nl-S 2 )*] 

Q _ N 2 2 (Ni-S 2 )Q 3 +NUN 2 2 -S 2 ) tanh [iks 2 (Nl- S 2 )^] 
^ 2 NI(N 2 -S 2 )+N 2 2 (m-S 2 )Qz tanh [iks 2 (N\- S 2 )*] 

and so on. Q 2 , Q±, Q 5 . . . are obtained by cyclic permutation of indices. It should be 
noted, however, for M discrete strata that Q M =^ since effectively Sm=°° • The resultant 
Hertz vector for the air space 0<s<A is then formally given by eq (3.13) with the more general 
meaning now attached to the ionosphere reflection coefficient. In the general case, the rigorous 
evaluation of the integrals would be extremely involved. However, using arguments similar 
to those for the homogeneous ionosphere, the field may be approximated as a sum of residues 
evaluated at the poles of the integrand. Thus the contributions from the branch points are 
again neglected since for finitely conducting layers they correspond to heavily damped waves. 
Therefore, the residue series formula given by eq (3.15) is also applicable if the reflection co- 

198 





r 


1 








1 
\ 


5 2 




N 2 




S , 


f 
1 




N, 


t 


f 




h 

i 















Figure 10. Stratified model for ionosphere. 



efficient Ri(C n ) is replaced by Ri(C n ). The modal equation now reads 

~Ri(C n )R g (C n ) exp (—2ikhC n ) = exp (-i&rw) (13.7) 

where n takes integral values. 

A numerical treatment and an application for the special case of a two-layer ionosphere 
has been carried out and reported in the literature [21]. There is no intrinsic difficulty in 
extending such calculations to an indefinitely large number of such layers each with infinitesimal 
thickness. For finitely conducting strata such a process converges and leads to an adequate 
representation for a continuous refractive index profile. 

A great simplification to the formulas for a stratified ionosphere is effected if the refractive 
indices for all layers are large. For example, if |iVi|, \N 2 \ . . . |AT M | ->->l, then 

^ N.Q.+ N, tanh flfexJVQ 
^ 1 ^N 2 +N 1 Q 2 tanh (iks.NJ K 6t * } 

n ^ N 2 Q 3 +N, tanh (iks 2 N 2 ) 

^ 2= N z +N 2 Q d tanh {iks 2 N 2 ) {i6 * } 

and so on. Thus to this approximation, Qi does not depend on the angle of incidence or the 
factor C. In this case, the modal equation simplifies to 

khCn^wn+iAlCn (13.10) 

where 

A=4+|- (13.11) 

Regarding A/C n as a small quantity, this can be solved to give 

s-KS)?-^ 1 -©']" 4 < i3 - i2 » 

where e =l, e n =2(n^0). This is valid if \Akh\ <<1 and |A| <<M [1 — (irn/kh) 2 ]. Thus, at 
extremely low frequencies and for highly conducting layers, the propagation factor S n is 
expressible in a relatively simple form. 

The special case of a two layer ionosphere was considered in some detail in a previous paper 
[21]. Such a model was sufficient to explain the variation of the observed attenuation rate for 
frequencies of the order of 500 cps to 15 kc. 

Exponential Profiles 

At the extremely low frequencies it was seen that for a stratified model of the ionosphere, 
the factor Q does not depend on C or S. In fact, it is not difficult to show that the surface 
impedance Z at 2 — h looking outwards is given by 

Z=rioQi/Ni where r; ^1207r. 
199 



From this viewpoint is becomes quite easy to write down expressions for Qi for other profiles. 
For example, if the refractive index varies in an exponential fashion relatively simple formulas 
are obtained. Two cases which could be considered are 

iV(z) = 1.0for 0<3<A, 

N{z)^Nexp [zf( z -k)/l] for z>h, 

where the (— ) sign corresponds to a refractive index decreasing with height and the (+) sign 
corresponds to a refractive index increasing with height. In the above, Z is a scale factor; for 
example, at within a distance / above the lower edge of the ionosphere, the refractive index has 
changed from N to N/e or Ne for the two respective cases. 

Within the layers of the ionosphere propagation is vertically upwards and thus a compo- 
nent E of the electric field satisfies 



&E 
dz 



^ 2 +k 2 N 2 {z)E=0 for z>h. 



Solutions of this equation for the exponential form of N 2 (z) are 

const X/o (ikNle- (z ~ h)/r ) 
E= (13.13) 

ooDstXX , o(ifcM«+ c, -* )/l ) 

for the two respective cases, where I and K are modified Bessel functions. The transverse 
component of the magnetic field is then found from Maxwell's equations, e.g., inooH=()E/dz. 
The surface impedance Z at the lower edge of this model of the ionosphere is then defined by 

Z=E/H] 2=h . 

For the case when the refractive index decreases with height 

Hs<™=HS (1314 > 

and when the refractive index increases with height, 

For low frequencies, ikNl^-yji x with sc=|iV| kl. The arguments of modified Bessel functions 
are thus proportional to -yji. Numerical values are shown in the table for certain real values of 
x. In both_these cases, it may be observed that as the scale length I approaches infinity, Z 
becomes rj /N as it should. 

The simplicity of the above formulas for the exponential profiles is due to the inherent 
assumption that the refractive index N{z) is large compared to unity for z^>h. When this 
condition is violated, such as it would be at frequencies above several kilocycles per second, the 
solution is not expressible in closed form, except for horizontal polarization [32] which is really 
only of academic interest at vlf . Nevertheless for calculation of attenuation rates at extremely- 
low-frequencies (less than 1 kc, say), the exponential models find direct application. For ex- 
ample, if Q=l the homogeneous ionosphere is regained and the attenuation rate {—1c Im S ) is 
proportional to ^ ; on the other hand for an exponential profile with N{z) increasing upwards, 
the attenuation rate increases with frequency more rapidly as suggested by experimental data. 
In fact, it is quite easy to see, for both decreasing and increasing exponential profiles, that 

Attenuation rate for exponential model __ ^ ,^, . />__ ~\ 
Attenuation rate for homogeneous model A '^' ' \4 ° / 

200 



Table 1. Selected numerical values of the factor Q 





Q- 7o < 


y'i x) 

V7i) 


Q- Ko 

A', 




J 


\Q\ 


arg Q 


IPI 


arg Q 


0.2 


10. 000 


-44°43' 


0. 3854 


23°58' 


0.5 


4. 003 


-43° 13' 


0. 5880 


17°16' 


1.0 


2. 026 


-37°55' 


0.7344 


11°59' 


1.5 


1.417 


-29°52' 


0.8047 


9°15' 


2.0 


1.180 


-20° 59' 


0.8469 


7°28' 


2.5 


1.100 


-13°44' 


0.8728 


6°20' 


3.0 


1.083 


-9°09' 


0.8919 


5°28' 


4.0 


1.080 


-5°42' 


0.9169 


4°18' 


5.0 


1.073 


-4°37' 


0.9325 


3°33' 


6.0 


1.060 


-3°49' 


0. 9433 


3°02' 


7.0 


1.051 


-3° 19' 


0.9511 


2°37' 


8.0 


1.045 


-2°47' 


0. 9570 


2°16' 


9.0 


1.040 


-2°26 / 


0. 9618 


2°06' 


10. 


1.035 


-2°11' 


0. 9654 


1°54' 


0.1 






0.25518 


27°22' 


0.3 






0.4719 


21°08' 


0.7 






0. 6622 


14°38' 



The right-hand side of the above equation is a function of the quantity x=|iV|Z^(o3 r /w)i/ where 
co = cr/€o_is the conductivity at the lower edge of the ionosphere and I is the vertical distance [in 
which N changes by a factor 2.718. 



The author thanks Mr. Kenneth Spies for his careful reading of the manuscript and [Mrs. 
Ami Fails for her typing and assistance in preparing the bibliography. 

14. References 



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des blitze, II Nuovo cimento IX (1952). 
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Proc. IRE, 45, 772 (1957). 
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Radiotekh. i Elektron., 1, 281 (1956). 
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3, paper 25, Boulder, Colo., (1957). 
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532053— 60- 



201 



[19] J. R. Wait and H. H. Howe. The waveguide mode theory of vlf ionospheric propagation, Proc. IRE 

45, (1957). 
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durch blitzentladungen, Z. angew. Phys. 9, 373 (1957). 
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(1952). 

15. Additional References 

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202: 



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203 



The following two papers have just appeared: 

Barron, D. W., The "waveguide mode" theory of radio wave propagation when the ionosphere is not sharply 

bounded, Phil. Mag. 50, 1068 (1959). (This is a continuation of early unpublished work of Budden.) 
Poverlein, H., Lang-und Langstwellausbreitung, Fortschr. der Hochfrequenztech. 4, 47 (1959). (Many 

additional references are found in this excellent review article.) 

Boulder, Colo. (Paper 64D2-49) 



204